\documentclass[11pt,a4paper,reqno]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=2.5cm]{geometry} \usepackage{amsmath,amssymb,mathtools,mathrsfs} \usepackage{microtype,booktabs,longtable,array,enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{pdftitle={The Humphreys support conjecture}, pdfauthor={Qiyue Tang}, pdfsubject={Support varieties of tilting modules in characteristic p greater than h}} \linespread{1.10} \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\kk}{\Bbbk} \newcommand{\CC}{\mathbb C} \newcommand{\QQ}{\mathbb Q} \newcommand{\ZZ}{\mathbb Z} \newcommand{\cN}{\mathcal N} \newcommand{\cO}{\mathcal O} \newcommand{\cC}{\mathcal C} \newcommand{\cH}{\mathcal H} \newcommand{\cT}{\mathcal T} \newcommand{\sgn}{\operatorname{sgn}} \DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Irr}{Irr} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\IC}{IC} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\rk}{rank} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\Spin}{Spin} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\Gr}{Gr} \DeclareMathOperator{\OG}{OG} \newcommand{\id}{\mathrm{id}} \newcommand{\eps}{\varepsilon} \title[The Humphreys support conjecture]{The Humphreys support conjecture} \author{Qiyue Tang} \address{Zhili College, Tsinghua University, Beijing 100084, China} \email{tangqy24@mails.tsinghua.edu.cn} \date{November 12, 2026} \subjclass[2020]{Primary 20G05; Secondary 20G10, 20C08, 14F08} \keywords{Tilting modules, support varieties, affine Weyl groups, Kazhdan--Lusztig cells, Springer correspondence} \begin{document} \begin{abstract} We prove the Humphreys support conjecture for connected reductive groups over algebraically closed fields of characteristic \(p>h\). \end{abstract} \maketitle \enlargethispage{2pt} \tableofcontents \clearpage % Begin sections/introduction.tex \section{Introduction} \subsection{The support formula} Let \(\kk\) be an algebraically closed field of characteristic \(p>0\), and let \(G\) be a connected reductive group over \(\kk\). Fix a maximal torus and a Borel subgroup. Write \(X\) for the character lattice of the torus, \(\Phi\subset X\) for the root system, \(\Phi^+\) for the chosen positive roots, and \(X^+\) for the dominant weights. A finite-dimensional \(G\)-module is tilting if it has a filtration by Weyl modules and a filtration by dual Weyl modules. The indecomposable tilting module of highest weight \(\lambda\in X^+\) is denoted by \(T_p(\lambda)\). Put \(h=\max h_i\), where the \(h_i\) are the Coxeter numbers of the irreducible components of \(\Phi\); set \(h=1\) for a torus. Throughout the paper we assume \[ p>h. \] Let \(G_1\) be the kernel of the relative Frobenius morphism \(G\to G^{(1)}\), and let \(\cN_{\kk}\) be the nilpotent cone. For a finite-dimensional \(G\)-module \(M\), its full cohomological support is \begin{equation}\label{intro:fullsupport} V_{G_1}(M) =\Supp_{H^{2\bullet}(G_1,\kk)} \Ext^\bullet_{G_1}(M,M). \end{equation} The cohomology ring is the coordinate ring of the Frobenius twist \(\cN_{\kk}^{(1)}\); see \cite[Lemma~8.1]{AHR}. Using the Frobenius homeomorphism, we regard the support as a closed \(G\)-stable subset of \(\cN_{\kk}\). Supports throughout this paper are closed sets. Let \(W\) be the finite Weyl group, let \(Q=\ZZ\Phi\), and put \(\rho=\frac12\sum_{\alpha\in\Phi^+}\alpha\). The affine and extended affine Weyl groups and their scaled dot action are \[ W_{\mathrm{aff}}=Q\rtimes W,\qquad W_{\mathrm{ext}}=X\rtimes W,\qquad (t_\nu w)\bullet_p\lambda=p\nu+w(\lambda+\rho)-\rho. \] Here \(t_\nu\) denotes translation by \(\nu\) before scaling by \(p\). The fundamental alcove is \[ \mathscr A_0= \{\lambda\in X\otimes_\ZZ\mathbb R: 0<\langle\lambda+\rho,\alpha^\vee\rangle

h\). For an alcove described by \(m_\alpha p<\langle\lambda+\rho,\alpha^\vee\rangle <(m_\alpha+1)p\), its \emph{lower closure} is obtained by replacing the left inequalities by weak inequalities. Each weight lies in the lower closure of a unique alcove. At a wall, this is the alcove containing \(\lambda+\epsilon\eta\) for sufficiently small \(\epsilon>0\), where \(\eta\) is strictly dominant. Lusztig's correspondence attaches a nilpotent orbit \(\cO_w\) to each ordinary affine two-sided cell \cite{LusztigIV}. If the lower-closure alcove of \(\lambda\) is \(w\bullet_p\mathscr A_0\), put \(\cO_\lambda=\cO_w\). Length-zero elements and central characters preserve this orbit label. For a torus the label is the zero orbit. \begin{theorem}\label{main} For every connected reductive group \(G\), every prime \(p>h\), and every dominant weight \(\lambda\), \[ V_{G_1}\bigl(T_p(\lambda)\bigr) =\overline{\cO_\lambda}. \] In particular, if \(w\) is a minimal representative in \(W\backslash W_{\mathrm{ext}}\), then \[ V_{G_1}\bigl(T_p(w\bullet_p0)\bigr)=\overline{\cO_w}. \] \end{theorem} \subsection{Background and previous results} The support theory for the first Frobenius kernel is rooted in the cohomology of restricted Lie algebras. Friedlander--Parshall developed support varieties for restricted enveloping algebras in 1986 \cite{FP}. For a rational representation of a reductive group, restriction to its first Frobenius kernel places the module in this setting. The resulting support is stable under conjugation, so nilpotent orbits provide its natural geometric constituents. The orbit label in the theorem comes from Lusztig's study of cells in affine Weyl groups. In \emph{Cells in affine Weyl groups. IV} \cite{LusztigIV}, he constructed the bijection between two-sided cells and unipotent classes in the corresponding complex reductive group. Under the nilpotent--unipotent correspondence, these are nilpotent orbits. He also identified the value of the \(a\)-function on a cell with the dimension of the Springer fiber of its orbit. For a nilpotent element \(x\), the Springer fiber \(\mathcal B_x\) is the variety of Borel subgroups whose Lie algebras contain \(x\); write \(\mathcal B_{\cO}\) for a fiber with \(x\in\cO\). Lusztig's construction uses affine Hecke algebras, their asymptotic based rings, and restriction to finite parabolic Hecke algebras. These finite restrictions will also enter our proof. In 1997, Humphreys proposed the support formula in \cite[\S12]{Humphreys}, in a comparison of modular representations of semisimple groups with representations of their Lie algebras. Humphreys asked whether, for indecomposable tilting modules with regular highest weights, it is exactly the orbit closure determined by the affine-cell correspondence. Translation functors extend the formulation to the lower closures of alcoves. Cooper's 2010 work studied support formulae for singular weights and several infinite families of tilting modules \cite{Cooper}. Ginzburg--Kumar's calculation of small-quantum-group cohomology in 1993 provided the corresponding nilpotent-cone support theory for quantum groups at roots of unity \cite{GK}. The quantum tilting support formula was established first. In 1998, Ostrik computed the supports of tilting modules for quantum groups of type \(A\) \cite{Ostrik}. In 2006, Bezrukavnikov described the cohomology of quantum tilting modules by coherent sheaves on the Springer resolution and the nilpotent cone \cite{Bezrukavnikov}. His description uses the exotic \(t\)-structure and perverse-coherent sheaves, and identifies the relevant orbit through the affine-cell correspondence. It gives the quantum support formula in all types; the passage to full self-extension support is also explained in \cite[\S8.4]{AHR}. This quantum formula provides the characteristic-zero comparison used below. In 2026, Coulembier proved naturality of small-quantum-group support under restriction to Levi subalgebras for finite-dimensional type-\(1\) modules extending to Lusztig's quantum group, including tilting modules \cite[Theorem~4.2.1]{Coulembier}. For algebraic groups in positive characteristic, Hardesty proved the conjecture in type \(A\) in 2018 \cite{Hardesty}. His proof uses the partition description of affine weight cells to obtain an upper bound, and comparison with quantum tilting modules to obtain the opposite inclusion. The resulting formula holds throughout \(p>h\); see \cite[Theorem~1.4.1]{Hardesty} and \cite[\S8.2]{AHR}. In 2019, Achar--Hardesty--Riche proved, for every root system and every \(p>h\), the lower bound \[ \overline{\cO_\lambda} \subseteq V_{G_1}(T_p(\lambda)). \] They proved equality when \(p\) is sufficiently large, with a bound depending on the root system \cite[Theorem~9.3]{AHR}. Their argument develops integral exotic parity objects and compares their coherent supports under specialization. Tilting characters in characteristic \(p\) are governed by the \(p\)-canonical basis. Parity sheaves provide its geometric construction \cite{JMW}, and the character theorem of Achar--Makisumi--Riche--Williamson identifies the antispherical \(p\)-canonical basis with modular tilting characters for \(p>h\) \cite{AMRW}. Ordinary canonical elements give quantum tilting characters. Consequently a character-preserving quantum lift of a modular indecomposable tilting module can have several indecomposable summands. Finite \(p\)-cells have also been studied directly. The subregular and submaximal cells in finite classical Weyl groups are described in the 2026 work of Miemietz--Roth--Tubbenhauer \cite[Theorems~3.11 and~4.10]{MRT}. Our finite constructions use ordinary cell bounds and particular \(p\)-independent objects inside the finite Hecke categories. In their 2024 papers, Achar--Hardesty constructed co-\(t\)-structures on coherent derived categories of the nilpotent cone and parabolic Springer resolutions \cite{AHCoT}. The nonzero pushforwards of exotic parity objects are indecomposable objects in the corresponding cohearts. Their later construction by nilpotent orbits and silting complexes identifies the support of these objects and proves the relative Humphreys conjecture for \(p>h\) \cite[Theorem~8.1 and Remark~8.2]{AH}. This determines the support of \(H^\bullet(G_1,T)\) for the tilting modules with nonzero cohomology. In our proof, a coevaluation summand of \(T\otimes T^*\) relates that relative support to the full support \eqref{intro:fullsupport}. The ICM 2026 account of Achar--Riche records these developments \cite[\S3 and Tables~1--2]{ARICM}. Its Conjecture~3.3 is the traditional support formula for weights in the extended affine orbit of \(0\), and Remark~3.4 explains its equivalence with the formulation for arbitrary dominant weights. The passage from relative to full support requires a compatibility with ordinary cells \cite[Remark~3.11(2)]{ARICM}. Theorem~\ref{main} proves the traditional support formula throughout \(p>h\). \subsection{Character and Springer conventions} In the proof up to the final reduction in Section~\ref{sec:conclusion}, \(G\) is assumed to be semisimple and simply connected. We use the positive Kazhdan--Lusztig normalization \(C_s=H_s+v\), where \(H_s\) is the standard Hecke generator; thus \(C_s(1)=1+s\). The standard and ordinary canonical Hecke bases are denoted by \(H_w\) and \(C_w\), respectively. Write \(\cH(W)\) and \(\cH(W_{\mathrm{aff}})\) for the corresponding Hecke algebras. The antispherical module is \[ \mathcal M_{\mathrm{asph}} =\operatorname{sgn}\otimes_{\cH(W)}\cH(W_{\mathrm{aff}}), \] where \(H_s\) acts on \(\operatorname{sgn}\) by \(-v\). The ordinary and \(p\)-canonical bases are denoted by \(N_w\) and \({}^pN_w\), indexed by the minimal representatives of \(W\backslash W_{\mathrm{aff}}\). Under the regular-block character identification, their specialization at \(v=1\) gives quantum and modular tilting characters, respectively \cite{AHR,AMRW}. The ordinary two-sided preorder is oriented so that \(x\leq_{LR}w\) implies \(\cO_x\subset\overline{\cO_w}\). A \emph{downset} is a union of nilpotent orbits closed under taking orbit closures. Let \(G_{\CC}\) be the connected complex reductive group with the same root datum as \(G\). We use the ordinary Springer convention in which the regular nilpotent orbit corresponds to the trivial representation and the zero orbit to the determinant representation. For a finite parabolic Weyl group \(H\), a standard Springer label \(F\) acts in the positive Hecke model as \(F^+=\det_H\otimes F\). We compare orbit labels in characteristic zero and characteristic \(p\) by their common good-characteristic parametrization. Write \(\iota\) for the resulting correspondence on unions of orbit closures; it preserves the closure order \cite[\S4.1]{AHR}. We use the same orbit notation over \(\CC\) and \(\kk\); the ambient support variety specifies the field. \subsection{Organization of the proof} Let \(q\) be a primitive \(p\)-th root of unity, and let \(u_q\) be the small quantum group for the same root system. For a modular tilting module \(M\), let \(Q(M)\) be the quantum tilting module having the same character, with all its indecomposable summands retained. We first prove \begin{equation}\label{intro:lift} V_{G_1}(M)=\iota V_{u_q}\bigl(Q(M)\bigr). \end{equation} Here quantum support is defined by self-extensions over \(u_q\). The proof uses a coevaluation summand, the relative support theorem, and specialization of an integral exotic parity object at the specified prime. Equation~\eqref{intro:lift} reduces the theorem to an upper bound on the ordinary quantum summands of \(Q(T_p(\lambda))\). For a downset \(\cC\), let \(J^q_{\cC}\) be the complex span of quantum tilting characters supported in \(\cC\), and let \(J^+_{\cC}\) be the span of modular tilting characters whose quantum lifts are supported there. We prove \begin{equation}\label{intro:character-equality} J^+_{\cC}=J^q_{\cC}. \end{equation} Tensor operations and linkage projections generate a coefficient operator algebra. Its stable character spaces correspond to Weyl-stable ideals in the group algebra \(\CC[X]\), with an additional finite-center stability condition. The resulting correspondence commutes with completion. At a semisimple torus point, let \(M\subset W\) be its stabilizer. The ordinary coefficient ideal is contained in the sum of the polynomial \(M\)-isotypic ideals whose local Springer orbits, viewed in the ambient nilpotent cone, lie in \(\cC\). A geometric comparison at Hecke parameter \(1\) proves this containment. Kato's canonical trace filtration shows that the sum is generated by the lowest polynomial copies of the Springer representations for trivial local systems. Finite parity objects and tensor operations supply these generators with the required support bounds. In classical types the construction is uniform in the rank. For the exceptional groups of rank at most six, finite \(p\)-canonical calculations provide the sources. In types \(E_7\) and \(E_8\), selected sources and exact finite-group traces give the local generators. The last \(E_8/A_7\) source is obtained from an integral calculation for a trivector double cover. The completed ideals of these finite constructions are constant on each root-stabilizer stratum. A boundary argument therefore extends the full-rank calculations to the whole torus. Finally, highest-weight triangularity gives the original label \(\cO_\lambda\), and translation and central isogenies give the statement for all weights and all connected reductive groups. Appendix~\ref{app:certificates} contains the finite tables and their verification rules. % End sections/introduction.tex % Begin sections/whole_lift.tex \section{The whole quantum lift} Let \(q\) be a primitive \(p\)-th root of unity, let \(U_q\) be Lusztig's quantum group, and let \(u_q\) be its small quantum group. For \(\lambda\in X^+\), let \(T_q(\lambda)\) denote the indecomposable quantum tilting module of highest weight \(\lambda\). We write \(\langle n\rangle\) for an internal grading shift and \([n]\) for a cohomological shift. A modular tilting module \(M\) has a unique quantum tilting module \(Q(M)\), up to isomorphism, with \(\ch Q(M)=\ch M\); see \cite[Remark~7.3]{AHR}. Uniqueness follows from the linear independence of indecomposable quantum tilting characters. In particular, it gives objectwise isomorphisms \begin{equation}\label{lift:character} \begin{split} Q(M\oplus N)&\cong Q(M)\oplus Q(N),\\ Q(M\otimes N)&\cong Q(M)\otimes Q(N),\\ Q(M^*)&\cong Q(M)^*. \end{split} \end{equation} These objectwise identities also preserve every direct-summand decomposition. \subsection{A coevaluation summand} Put \(R_G=H^{2\bullet}(G_1,\kk)\). For a finite-dimensional rational module \(M\), the annihilator of its self-extension module is the kernel of the central action \[ R_G\longrightarrow\Ext^\bullet_{G_1}(M,M). \] Denote this ideal by \(I_M\). Indeed, a class annihilating the self-extension module kills its identity in degree zero; conversely, the action on every extension is Yoneda composition with the action on that identity. \begin{lemma}\label{lift:anchor} Let \(T\) be an indecomposable tilting module. There are an indecomposable tilting summand \(D\) of \(T\otimes T^*\) and a map \(\delta:\kk\to D\) such that \(\delta\otimes1_T:T\to D\otimes T\) has a left inverse. Moreover, \[ I_T=I_D =\Ann_{R_G}H^\bullet(G_1,D) =\Ann_{R_G}(R_G\delta). \] \end{lemma} \begin{proof} Decompose \(T\otimes T^*=\bigoplus_iD_i\), and let \(\delta_i:\kk\to D_i\) be the components of coevaluation. Evaluation gives maps \(s_i:D_i\otimes T\to T\) with \[ \sum_i s_i(\delta_i\otimes1_T)=1_T. \] The finite-dimensional ring \(\End_G(T)\) is local. Its nonunits form an ideal, so at least one summand in this sum is invertible. Replacing the corresponding \(s_i\) by an invertible multiple gives \(s(\delta\otimes1_T)=1_T\). For \(a\in H^{2n}(G_1,\kk)\), write \(a_T\) for its central action on \(T\). Tensoring the class \(a\delta\in\Ext^{2n}_{G_1}(\kk,D)\) with \(T\) gives \[ s[2n]\circ\bigl((a\delta)\otimes1_T\bigr)=a_T, \] because \(s(\delta\otimes1_T)=1_T\). Thus \(a\delta=0\) implies \(a_T=0\). Conversely, \(a_T=0\) implies \(a_{T\otimes T^*}=a_T\otimes1_{T^*}=0\); restriction to the summand \(D\), followed by composition with \(\delta\), gives \(a\delta=0\). Hence \(I_T=\Ann(R_G\delta)\). The retractions \[ D\mid T\otimes T^*,\qquad T\mid D\otimes T \] give \(I_T=I_D\). Finally, \[ I_D\subseteq\Ann H^\bullet(G_1,D) \subseteq\Ann(R_G\delta)=I_T, \] which proves all equalities. \end{proof} The nonzero map \(\delta\) puts \(D\) in the principal linkage component. Write \(D=T_p(d\bullet_p0)\), where \(d\in W_{\mathrm{aff}}\) is minimal in \(Wd\). Since \(\Hom_G(\kk,D)\ne0\), the element \(d\) is also minimal in \(dW\), and hence in \(WdW\), by \cite[Lemmas~8.7--8.8]{AHR}. The relative support theorem and its ordinary-cell identification \cite[Theorem~8.1 and Remark~8.2]{AH}, together with Lemma~\ref{lift:anchor}, give \begin{equation}\label{lift:anchororbit} V_{G_1}(T)=V_{G_1}(D) =\iota\overline{\cO_d}. \end{equation} \subsection{Specialization of the entire parity object} At the fixed prime \(p>h\), choose a Bott--Samelson object containing the exotic parity object of the given principal-block tilting. Its degree-zero endomorphism algebra is finite-dimensional over \(\mathbb F_p\) and splits over a finite extension. Over such an extension choose an indecomposable summand whose base change is the given parity object. Lift its primitive idempotent over a complete discrete valuation ring \(\mathbb O\), finite over \(\ZZ_p\), to obtain a summand of the integral Bott--Samelson object. Choose embeddings of its fraction field into \(\CC\) and its residue field into \(\kk\). Extension along the latter recovers an isomorphic geometric object. Its standard-Hom modules are finite free and commute with base change. Simultaneous integral representatives for nilpotent orbits can be chosen over such a ring. These assertions follow from \cite[Corollaries~3.4 and~3.6, the proof of Lemma~3.11, Lemma~4.5, and \S4.4]{AHR}. \begin{lemma}\label{lift:specialization} For the summand \(D\) of Lemma~\ref{lift:anchor}, \[ \iota V_{u_q}\bigl(Q(D)\bigr) \subseteq V_{G_1}(D). \] \end{lemma} \begin{proof} Let \(\mathcal E_{\mathbb O}\) be an integral exotic parity lift of the object corresponding to \(D\). Its characteristic-zero fiber is a finite direct sum of indecomposable exotic parity objects, with diagonal grading shifts. For every standard object \(\Delta_{\mathbb O}(\mu)\), the groups \[ \Hom\bigl(\Delta_{\mathbb O}(\mu), \mathcal E_{\mathbb O}\langle n\rangle[n]\bigr) \] are free and compatible with both fibers. Indeed this holds for the Bott--Samelson object and passes to the image of its lifted idempotent. The degrading equivalences identify the finite sums of these ranks with the corresponding standard-filtration multiplicities. Thus the entire generic fiber has quantum character \(\ch D\), and its quantum image is \(Q(D)\). The principal root-lattice component satisfies the equivariance condition in the quantum comparison \cite[Proposition~9.1 and Remark~9.2]{AHR}. Write \(\widetilde\pi\) for the Springer map over \(\mathbb O\). After forgetting equivariance, consider \[ \mathcal K_{\mathbb O} =R\widetilde\pi_* R\mathcal Hom(\mathcal E_{\mathbb O},\mathcal E_{\mathbb O}). \] The Springer scheme over \(\mathbb O\) is regular, so \(\mathcal E_{\mathbb O}\) is perfect. The map is proper and \(\mathcal K_{\mathbb O}\) is a bounded coherent complex. For a simultaneous integral nilpotent representative \(x:\Spec\mathbb O\to\cN_{\mathbb O}\), its derived pullback is a bounded-above pseudo-coherent complex over \(\mathbb O\), with finitely generated cohomology in each degree. The integral Springer scheme and nilpotent cone are flat over \(\mathbb O\), so their scalar-change squares are Tor-independent. Proper base change identifies the generic and special fibers of this pullback with the pullbacks of the respective self-Hom pushforwards. If its special fiber were zero, derived Nakayama over \(\mathbb O\) would make the pullback zero, and hence its generic fiber zero. Therefore every nonzero generic fiber has nonzero special fiber. The self-Hom of each generic indecomposable summand is a direct summand of the self-Hom of the whole generic object. Hence the support of every such summand specializes into the special self-extension support. The modular and quantum coherent support comparisons identify these supports with the two sides of the asserted inclusion. \end{proof} \begin{theorem}\label{lift:whole} For every tilting \(G\)-module \(M\), \[ V_{G_1}(M) =\iota V_{u_q}\bigl(Q(M)\bigr). \] \end{theorem} \begin{proof} First let \(M=T\) be indecomposable, and choose \(D\) as above. The top quantum tilting \(T_q(d\bullet_p0)\) occurs in \(Q(D)\) with multiplicity one. Quantum support theory gives \[ \overline{\cO_d}\subseteq V_{u_q}(Q(D)). \] Together with Lemma~\ref{lift:specialization} and \eqref{lift:anchororbit}, this proves \(\iota V_{u_q}(Q(D))=V_{G_1}(T)\). The objectwise identities \eqref{lift:character} transfer the two retractions in Lemma~\ref{lift:anchor} to quantum tilting modules: \[ Q(D)\mid Q(T)\otimes Q(T)^*,\qquad Q(T)\mid Q(D)\otimes Q(T). \] For quantum modules \(A,B\), the central cohomology action on \(A\otimes B\) is obtained by tensoring the action on \(A\) with \(1_B\), so \(V_{u_q}(A\otimes B)\subseteq V_{u_q}(A)\). Apply this inclusion and summand monotonicity to the two retractions to obtain \(V_{u_q}(Q(T))=V_{u_q}(Q(D))\). For arbitrary \(M\), take its finite decomposition into indecomposable tiltings and use the direct-sum formula for support. \end{proof} \begin{corollary}\label{lift:transitions} If \[ {}^pN_w=\sum_x m_{x,w}(v)N_x, \] then \[ V_{G_1}\bigl(T_p(w\bullet_p0)\bigr) =\bigcup_{m_{x,w}\ne0}\iota\overline{\cO_x}. \] In particular, it contains \(\iota\overline{\cO_w}\). \end{corollary} \begin{proof} The Laurent polynomials \(m_{x,w}\) have nonnegative integral coefficients, and \(m_{w,w}=1\). Their values at \(1\) are the quantum summand multiplicities in \(Q(T_p(w\bullet_p0))\). Apply Theorem~\ref{lift:whole} and the quantum support formula. \end{proof} % End sections/whole_lift.tex % Begin sections/operators.tex \section{Actual character operators and coefficient ideals} \label{sec:operators} All vector spaces and algebras in this section are over \(\CC\). The characteristic \(p\) remains the fixed modular characteristic. Put \[ R=\CC[X],\qquad A=R^W,\qquad \Delta=\sum_{w\in W}\det(w)e^{w\rho},\qquad \psi_p(e^\nu)=e^{p\nu}. \] Multiplication by \(\Delta\) identifies the character ring \(A\) with the anti-invariants \(R^{\det}\). An \emph{actual character operation} is tensoring with a modular tilting module, projection to a linkage component, or a composite of these operations. We take their complex linear span only after they have been applied to actual modules. \subsection{The large operator algebra} Let \(\mathscr B\) be the operator algebra on \(R^{\det}\) generated by multiplication by \(A\) and by the projections onto \(W\)-orbits of residues in \(X/pQ\). The shift by \(\rho\) has already been incorporated in the numerator model. These are exactly the available linkage projections, with a projection understood as the sum of categorical block projections when necessary. Since tilting characters form a basis of \(A\), multiplication by any element of \(A\) is a linear combination of actual tensor operations. Set \[ R_0=\CC[pQ],\qquad D_0=R\rtimes\Hom(X/pQ,\CC^\times),\qquad E_0=D_0^W. \] Fourier projections onto individual residue classes, followed by multiplication by Laurent monomials, give all matrix units. Consequently \begin{equation}\label{op:matrix} D_0=\End_{R_0}(R). \end{equation} The action of \(E_0\) on \(R^{\det}\) is faithful. To see this, let \(e_{\det}=|W|^{-1}\sum_w\det(w)w\) in \(D_0\rtimes W\). The corner \(e_{\det}(D_0\rtimes W)e_{\det}\) is identified with \(E_0\) by \(a\mapsto ae_{\det}\). Distinct automorphisms of the fraction field of \(R\) are linearly independent over that field, so the crossed product acts faithfully on \(R\). If \(a\in E_0\) kills \(R^{\det}\), then \(ae_{\det}\) kills \(R\); its identity-\(W\) coefficient gives \(a=0\). In particular, equalities of the stated character operators are equalities in \(E_0\). \begin{proposition}\label{op:full} For \(p>h\), the actual operator algebra is \(E_0\). \end{proposition} \begin{proof} Let \(f\) be the principal linkage projector. Since \(0\) lies in the fundamental alcove, the weights \(w\rho\) are distinct modulo \(pQ\), and \[ fR=\bigoplus_{w\in W}R_0e^{w\rho}. \] This is the left regular module of \(R_0\rtimes W\). It follows that \begin{equation}\label{op:corner} fE_0f=\End_{R_0\rtimes W}(fR) \cong(R_0\rtimes W)^{\mathrm{op}}. \end{equation} Contracting \(pQ\) to \(Q\), this is the opposite affine group algebra. On anti-invariants the corresponding map is \[ g\longmapsto \Delta^{-1}\sum_{w\in W}\det(w)e^{w\rho}\psi_p(wg), \qquad g\in\CC[Q]. \] Under the antispherical identification, right translation by \(\nu\) acts by multiplication by \(e^\nu\), while a finite Weyl element acts by its determinant times the inverse polynomial action. The wall-crossing formula \cite[\S7.2, equation~(7.1)]{AHR} is therefore precisely \(\Theta_s=1+s\). Subtracting the identity obtains every affine simple reflection, and these generate the affine group algebra. Thus \(f\mathscr Bf=fE_0f\). It remains to prove that \(f\) is full in \(\mathscr B\). Represent each linkage class by an integral point \(\mu\) in the closed fundamental alcove, and let \(W_J\) be its finite affine stabilizer. The representative \(\mu\) need not be dominant. Translation out of the \(\mu\)-block and back has character \begin{equation}\label{op:translation} T_0^\mu T_\mu^0=|W_J|\id. \end{equation} Indeed, for a standard character with strongly minimal double-coset label \(y\), translation out has the standard characters with labels \(yz\), \(z\in W_J\), each once; translation back sends each to the original standard character. This is the translation principle \cite[\S2.5(a),(b), following equation~(2.4)]{ARICM}. On characters, translation is multiplication by the character of a finite-dimensional module, preceded and followed by linkage projections. That character belongs to \(A\), which is spanned by tilting characters. Its translation operator therefore lies in \(\mathscr B\). Hence, writing \(P_\mu\) for the linkage projector, \[ P_\mu=|W_J|^{-1}T_0^\mu fT_\mu^0 \in\mathscr Bf\mathscr B. \] There are finitely many residue orbits. Summing gives \(1=\sum_i b_i f c_i\), with \(b_i,c_i\in\mathscr B\). For any \(x\in E_0\), \[ x=\sum_{i,j}b_i(f c_i x b_j f)c_j. \] The middle factor belongs to \(fE_0f=f\mathscr Bf\). Thus \(x\in\mathscr B\), as required. \end{proof} \subsection{The small determinant corner} Put \(R_p=\CC[pX]\) and \[ D=\End_{R_p}(R) =R\rtimes\Hom(X/pX,\CC^\times),\qquad \mathscr S=D\rtimes W. \] Let \[ Z=\Hom(X/Q,\CC^\times), \qquad \tau_z(e^\lambda)=z(\lambda)e^\lambda \quad(z\in Z). \] Every prime divisor of \(|X/Q|\) is at most \(h\), so \(p\) is prime to \(|Z|\). The subgroups \(\Hom(X/pX,\CC^\times)\) and \(Z\) of \(\Hom(X/pQ,\CC^\times)\) have coprime orders and their product has order \(|X/pQ|\). Consequently \begin{equation}\label{op:central-decomposition} \begin{aligned} \Hom(X/pQ,\CC^\times) &=\Hom(X/pX,\CC^\times)\times Z,\\ E_0&=D^W\rtimes Z. \end{aligned} \end{equation} For the second equality, first write \(D_0=D\rtimes Z\). The Weyl group acts trivially on \(X/Q\), so taking \(W\)-invariants coefficientwise in the \(Z\)-basis gives \((D\rtimes Z)^W=D^W\rtimes Z\). \begin{lemma}\label{op:rho} The weights \(w\rho\), \(w\in W\), are pairwise distinct modulo \(pX\). \end{lemma} \begin{proof} If \(w\rho-\rho=p\nu\), the extended affine element \(t_{-\nu}w\), acting with translation scale \(p\), fixes \(\rho\) in the unshifted fundamental alcove. It therefore belongs to the stabilizer \(\Omega\) of that alcove. The affine wall coordinates of \(\rho\) are \[ (p-h+1,1,\ldots,1). \] The affine coordinate is the unique coordinate greater than \(1\). An element of \(\Omega\) fixing \(\rho\) must fix the affine wall and its opposite vertex \(0\). It then has zero translation part; an alcove-preserving element of \(W\) is the identity. This proves the assertion in an irreducible root system, and products follow factor by factor. \end{proof} \begin{proposition}\label{op:coefficient} There is an inclusion-preserving bijection between character subspaces \(J\subset A\) stable under actual tensor operations and linkage projections, and ideals \(I\subset R\) stable under \(W\) and all central translations \(\tau_z\), \(z\in Z\). For each such subspace, \begin{equation}\label{op:coefficient-formula} \Delta J=(R\psi_p(I))^{\det}. \end{equation} In particular, this construction applies to the modular and quantum support subspaces. It is an equality of modules and ideals, including their completions. \end{proposition} \begin{proof} By Lemma~\ref{op:rho}, the module \(R\) contains the \(R_p\rtimes W\)-stable direct summand \[ \bigoplus_{w\in W}R_p e^{w\rho} \cong R_p\otimes_\CC\CC[W]. \] The right-hand side is the regular module of \(R_p\rtimes W\). Moreover \(R\) is projective over this skew group ring: it is free over \(R_p\), and a splitting can be averaged over \(W\). It is thus a projective generator, as is its determinant twist. Equivariant matrix Morita equivalence now says that \(e_{\det}\) is full in \(\mathscr S\). A submodule of the natural \(D=\End_{R_p}(R)\)-module \(R\) has the form \(R I_p\) for a unique ideal \(I_p\subset R_p\). It is \(\mathscr S\)-stable precisely when \(I_p\) is \(W\)-stable. Fullness of \(e_{\det}\) gives a bijection between these submodules and the submodules of \(R^{\det}\) for its corner \(D^W\). Transport \(I_p\) through \(\psi_p:R\to R_p\). This proves \eqref{op:coefficient-formula}, its uniqueness, and preservation of inclusions for \(D^W\)-submodules. For \(z\in Z\), one has \[ \tau_z(\Delta)=z(\rho)\Delta, \qquad \tau_z\psi_p=\psi_p\tau_{z^p}. \] Multiplication by the nonzero scalar \(z(\rho)\) preserves each subspace, and \(z\mapsto z^p\) permutes \(Z\). Thus the character submodule is \(Z\)-stable precisely when its coefficient ideal is \(Z\)-stable. Proposition~\ref{op:full} and \eqref{op:central-decomposition} now give the stated correspondence for all actual operators. All rings are Noetherian, and the modules in question are finite over \(R_p^W\). Hence the module and ideal identities commute with flat completion. \end{proof} \begin{remark}\label{op:extended} In the aggregate principal component modulo \(pX\), the coefficient map is \[ R\longrightarrow A,\qquad g\longmapsto \Delta^{-1}\sum_{w\in W}\det(w)e^{w\rho}\psi_p(wg). \] It is the extended antispherical polynomial model \[ \operatorname{sgn}\otimes_{\CC[W]}\CC[W_{\mathrm{ext}}] \longrightarrow R,\qquad 1\otimes t_\nu w\longmapsto\det(w)e^{w^{-1}\nu}. \] The length-zero components are realized by translation equivalences between regular linkage classes. Their character operators belong to \(\mathscr B\) by the same multiplication and linkage-projection formula. \end{remark} \subsection{The exact local tensor operation} Write \(\mathbb T=\Spec R\), put \(Y=\Hom_{\ZZ}(X,\ZZ)\), and write \(y^\lambda=e^\lambda(y)\). For \(y\in\mathbb T\), let \(M=W_y\) and \(S_y=\widehat{\mathcal O}_{\mathbb T,y}\). The centralizer of \(y\) in \(G_{\CC}\) is connected, since the ambient derived group is simply connected. Thus \(M\) is the reflection group of the roots \(\alpha\) with \(\alpha(y)=1\). Logarithmic coordinates \(u\in Y\otimes_{\ZZ}\CC\), defined by \[ e^\lambda(y\exp u) =y^\lambda\exp\langle\lambda,u\rangle, \] identify \(S_y\) with the power-series ring on this vector space, carrying the natural linear action of \(M\). If \(t^p=y\), then \(W_t\subseteq M\). Denote this subgroup by \(L_t\), and denote its linear reflection discriminant by \(\Delta_{L_t}\). For two operator-stable character families with coefficient ideals \(I,J\), let \(I\star J\) be the coefficient ideal of the operator closure of their actual tensor products. \begin{proposition}\label{op:tensor} In the completed local ring at \(y\), \begin{equation}\label{op:tensor-formula} \widehat{(I\star J)}_y =\sum_{t^p=y\;/M} S_y M\!\left[ \Delta_{L_t} (\widehat I_y:\Delta_{L_t})^{L_t} (\widehat J_y:\Delta_{L_t})^{L_t}\right]. \end{equation} The notation on the right means the ideal generated by the \(M\)-translates of all the indicated products. \end{proposition} \begin{proof} Let \(P\subset A\) be the character span of the tensor products. Both input character spaces are \(Z\)-stable by Proposition~\ref{op:coefficient}, and \(\tau_z\) is an algebra automorphism of \(A\). Hence \(P\), and therefore \(\Delta P\), is \(Z\)-stable. Since \(Z\) normalizes \(D^W\), the submodule \(D^W(\Delta P)\) is also \(Z\)-stable. Equation~\eqref{op:central-decomposition} gives \[ E_0(\Delta P)=D^W(\Delta P). \] It therefore suffices to compute the determinant-corner closure. Complete first over the \(W\)-orbit of \(y\) in the base torus. The characteristic-zero \(p\)-power map is finite etale and splits over the completion into its \(p\)-th-root local factors. In the factor at \(t\), an anti-invariant numerator is an \(L_t\)-anti-invariant element of \(\widehat I_y\). The Weyl numerator has the form \[ u_t\Delta_{L_t}, \] where \(u_t\) is an invertible \(L_t\)-invariant series: the vanishing roots give the discriminant, the other factors are units, and both numerator and discriminant transform by the same determinant character. Thus the local character module is \[ u_t^{-1}(\widehat I_y:\Delta_{L_t})^{L_t}. \] The analogous expression holds for \(J\). Multiplying characters and restoring one Weyl numerator gives the term in \eqref{op:tensor-formula}, up to the invariant unit \(u_t^{-1}\). That unit can be absorbed in either invariant colon module. The matrix algebra \(\End_{R_p}(R)\) extracts every coefficient component; after the etale splitting, this is the same as extraction of every root factor. Weyl saturation gives the outer \(M\)-span. Completion commutes with these products and images because all modules are finite over the invariant base. Their image submodules are adically closed, proving \eqref{op:tensor-formula}. \end{proof} \begin{corollary}\label{op:retention} Suppose a source at \(y\) contains a copy of an irreducible \(M\)-module \(E\). Let \(L=W_t\) occur for a \(p\)-th root of \(y\). If a second source contains \(\Delta_L\) and \(\Res^M_L E\) contains \(\det_L\), their tensor source retains a copy of \(E\). It retains the whole \(E\)-isotypic ideal if the first source contains that ideal. Its support is bounded by the intersection of the two source bounds. \end{corollary} \begin{proof} Choose a nonzero \(L\)-anti-invariant vector \(g\) in the given copy. It has the form \(\Delta_L h\) with \(h\in S_y^L\). The \(L\)-term of \eqref{op:tensor-formula} contains \(\Delta_L h\cdot1=g\). Its \(M\)-translates give the whole copy. Apply this argument in every degree for an entire isotypic ideal. The support assertion follows from tensor-product support for \(G_1\) and Theorem~\ref{lift:whole}. \end{proof} \subsection{The two support character spaces} For a closed nilpotent downset \(\cC\) in characteristic zero, define \[ \begin{split} J^q_{\cC}&=\operatorname{span}_{\CC} \{\ch T_q(\lambda):V_{u_q}(T_q(\lambda))\subseteq\cC\},\\ J^+_{\cC}&=\operatorname{span}_{\CC} \{\ch T_p(\lambda): V_{u_q}(Q(T_p(\lambda)))\subseteq\cC\}. \end{split} \] The whole-lift theorem also describes the second space using modular support in \(\iota\cC\). Both are coordinate subspaces in their respective tilting bases and are stable under actual operations. Let \(I_{\cC}\) and \(I^+_{\cC}\) be the coefficient ideals corresponding to \(J^q_{\cC}\) and \(J^+_{\cC}\), respectively. Then \begin{equation}\label{op:basic-inclusion} I^+_{\cC}\subseteq I_{\cC}. \end{equation} In the extended antispherical model, \(I_{\cC}\) is the coefficient image at \(v=1\) of the ordinary two-sided based ideal for \(\cC\). This follows by applying the quantum support theorem to its canonical antispherical basis and the coefficient identification of Remark~\ref{op:extended}. The next section describes this ideal near each semisimple point. % End sections/operators.tex % Begin sections/ordinary.tex \section{Ordinary cell ideals and Springer generators} \label{sec:ordinary} \subsection{Finite generation at parameter one} Let \(J\) be the asymptotic algebra of the equal-parameter affine Hecke algebra. We use its usual basis \(t_w\). Write \(C_xC_y=\sum_z h_{x,y,z}(v)C_z\). Lusztig's \(a\)-function is defined by \[ a(z)=\min\{n\geq0:v^n h_{x,y,z}(v)\in\ZZ[v] \text{ for all }x,y\}; \] its common value on a two-sided cell \(\mathbf c\) is denoted by \(a(\mathbf c)\). For a finite standard parabolic subgroup \(W_I\), let \(J^I\) be its finite asymptotic subalgebra. Over \(\CC\), Xi's finite-parabolic generation theorem \cite[Theorem~3.1]{Xi07} gives \begin{equation}\label{ord:xi} J_{\CC} =\sum_{I:\,W_I\text{ finite}}J_{\CC}J^I_{\CC}J_{\CC}. \end{equation} The statement applies to the affine and extended groups used here. The maximal extended algebra is the crossed product of the affine algebra by its length-zero group \(\Omega\). Conjugation by \(\Omega\) permutes the finite standard parabolics. The ideal generated by their affine subalgebras is therefore \(\Omega\)-stable. Equation~\eqref{ord:xi} in the crossed product implies the same equality in its identity homogeneous component. It then implies the equality in every intermediate extended group. The argument also applies to products. \begin{lemma}\label{ord:switch} Let \(B\) be a unital algebra with multiplication \(*\). Suppose a second associative multiplication \(\mu\) on its underlying vector space satisfies the two mixed associativity identities \[ \mu(x*y,z)=x*\mu(y,z),\qquad \mu(x,y*z)=\mu(x,y)*z. \] Writing \(e\) for the \(*\)-unit and \(d=\mu(e,e)\), one has \(\mu(x,y)=x*d*y\). Suppose unital subalgebras \(S_i\subset B\), with units \(e_i\), generate \(B\) as a \(*\)-ideal, and that \(d_i=e_i*d*e_i\) is invertible in \(S_i\). Then the same subspaces generate the whole vector space as a \(\mu\)-ideal. \end{lemma} \begin{proof} The first assertion follows by inserting \(e\) into the mixed identities. Let \(u_i=d_i^{-1}\) in \(S_i\). Since \(u_i=e_i*u_i=u_i*e_i\), \[ u_i*d*e_i*d*u_i=u_i*d_i*d_i*u_i=e_i. \] Choose a finite expression \(e=\sum_jb_j*e_{i_j}*c_j\). For every \(x\in B\), \[ \begin{aligned} &\sum_j\mu\bigl(x*b_j*u_{i_j}, \mu(e_{i_j},u_{i_j}*c_j)\bigr)\\ &\qquad=\sum_jx*b_j*u_{i_j}*d*e_{i_j}*d*u_{i_j}*c_j\\ &\qquad=\sum_jx*b_j*e_{i_j}*c_j=x. \end{aligned} \] This is an expression in the \(\mu\)-ideal generated by the \(S_i\). \end{proof} \begin{proposition}\label{ord:finite-generation} Let \(\mathscr U\) be a downset of ordinary affine two-sided cells. At \(v=1\), its based ideal is generated by its intersections with the finite standard parabolic Hecke algebras: \begin{equation}\label{ord:finite-generation-formula} \cH_{\mathscr U}(1) =\sum_{I:\,W_I\text{ finite}} \CC[W_{\mathrm{ext}}]\, \operatorname{span}_{\CC} \{C_x(1):x\in W_I\cap\mathscr U\}\, \CC[W_{\mathrm{ext}}]. \end{equation} Each finite intersection is a sum of complete matrix blocks of \(\CC[W_I]\). \end{proposition} \begin{proof} For a two-sided cell \(\mathbf c\), take its exact ordinary layer \(\cH_{\leq\mathbf c}/\cH_{<\mathbf c}\). Identify its basis with the asymptotic cell algebra \(J_{\mathbf c}\). The equal-parameter bimodule identity \cite[\S2.1(d)]{Xi07} gives \(\mu(x*y,z)=x*\mu(y,z)\) on each \(a\)-layer. The common anti-involution \(t_w\mapsto t_{w^{-1}}\), \(C_w\mapsto C_{w^{-1}}\) gives \(\mu(x,y*z)=\mu(x,y)*z\). P4 and P11 restrict these identities to the exact cell layer: every output cell is below the input cell, and equality of \(a\)-values forces the two cells to coincide. Thus Lemma~\ref{ord:switch} applies to \(J_{\mathbf c}\). Projecting \eqref{ord:xi} to \(J_{\mathbf c}\) gives generation by \[ S_{I,\mathbf c} =\operatorname{span}_{\CC} \{t_w:w\in W_I\cap\mathbf c\}. \] Finite-parabolic compatibility of the Kazhdan--Lusztig structure constants and equality of intrinsic and ambient \(a\)-values (property P12) identify this space with a direct sum of finite intrinsic cell algebras, with its own asymptotic unit \(e_{I,\mathbf c}\). At \(v=1\), the ordinary product on the same vector space is the quotient of the two ideals \[ \CC[W_I]\cap\cH_{\leq\mathbf c}(1),\qquad \CC[W_I]\cap\cH_{<\mathbf c}(1). \] The group algebra \(\CC[W_I]\) is semisimple. Its indicated ideal quotient therefore has a unit \(f_I\). If its product is \(x*d_I*y\), substituting the asymptotic unit in the ordinary unit identities gives \[ f_I*d_I=e_{I,\mathbf c}=d_I*f_I. \] Thus \(f_I\) is the required inverse in the specialized asymptotic corner. Lemma~\ref{ord:switch} now gives generation of each exact ordinary cell layer. Induction through a linear extension of the finite cell order, with lower cells first, gives \eqref{ord:finite-generation-formula}. Finally a two-sided ideal of a finite semisimple algebra is a sum of its full matrix blocks. \end{proof} \subsection{The orbit of a finite source} If a finite affine parabolic has projected group \(H\subset W\), a finite two-sided family has a unique special representation \(F_{\mathrm{sp}}\) in the standard Springer normalization. For an irreducible \(H\)-module \(F\), let \(b(F)\) be the least degree in which \(F\) occurs in the polynomial ring on the reflection representation of \(H\), or equivalently in its coinvariant algebra. \begin{proposition}\label{ord:finite-label} The ambient affine cell \(\mathbf c\) containing that finite family has orbit \[ \cO\bigl(j_H^W(F_{\mathrm{sp}})\bigr). \] Here \(j\) is induction truncated at the smallest coinvariant degree. It gives the Springer representation of a trivial local system. The Springer-fiber dimension of its orbit is \[ a(\mathbf c)=b(F_{\mathrm{sp}}). \] Before applying this rule, a family in the positive Hecke convention is twisted by \(\det_H\). \end{proposition} \begin{proof} First pass to the adjoint quotient, which has the same affine Coxeter group and finite standard parabolics. In this setting Lusztig defines the orbit by truncated induction in \cite[\S6.5]{LusztigIV}, and identifies its Springer fiber dimension with the finite \(a\)-value. The finite asymptotic module occurs in a simple ambient asymptotic module by \cite[\S5.4]{LusztigIV}. The multiplicity persists in its generic Hecke counterpart by \cite[\S5.6(b)]{LusztigIV}. Theorem~4.8 of that paper assigns the ambient cell and its \(a\)-value to the latter module. Its Lemma~7.2 puts the associated nilpotent element in the closure of the orbit defined in \S6.5, and puts it in that orbit when the Springer fiber dimensions agree. P12 supplies that equality. Sections~4.10--4.12 of \cite{LusztigIV} extend the classification from the adjoint case to the affine and extended groups of the given root datum. The length-zero group acts trivially on affine two-sided cells, and the orbit labels correspond under the central isogeny. The finite Weyl group, Springer fibers, and their trivial-local-system representations are unchanged. Thus the same truncated-induction label applies to the present group. The determinant twist is the positive-normalization convention already fixed above. \end{proof} \subsection{The local support containment} Let \(y\in\mathbb T\), \(M=W_y\), \(G_y=Z_{G_{\CC}}(y)\), and \(\mathfrak g_y=\operatorname{Lie}(G_y)\). Let \(\cC_y=\cC\cap\cN(\mathfrak g_y)\). For \(E\in\Irr(M)\), write \(\cO_M(E)\) for its ordinary Springer orbit. In this notation the local system is part of the label \(E\), although the support condition only uses its orbit. Define the completed Springer downset ideal \[ K_{\cC,y} =\sum_{\cO_M(E)\subseteq\cC_y}S_y\,S_y[E], \] where \(S_y[E]\) is the whole natural \(E\)-isotypic component. Write \(\mathfrak m_y\) for the maximal ideal of \(S_y\). \begin{proposition}\label{ord:containment} For every downset \(\cC\) and every \(y\in\mathbb T\), \[ \widehat I_{\cC,y} \subseteq K_{\cC,y}. \] Moreover, the completion of the ordinary based ideal for \(\cC\) is generated by constant matrix blocks in \(\operatorname{Mat}_{[W:M]}(S_y\rtimes M)\). Every selected block, expressed in the natural Springer convention, has a label \(E\) with \(\cO_M(E)\subseteq\cC_y\). \end{proposition} \begin{proof} Xi's theorem \cite[Theorem~1.5]{Xi11} identifies the rational ordinary based ideal with the image of equivariant coherent \(K\)-theory of the Steinberg correspondence supported over \(\cC\). Its complement in the Kazhdan--Lusztig basis is free over \(\QQ[v,v^{-1}]\). Consequently specialization at \(v=1\) is injective on this based ideal and commutes with the supported image. Put \(n=[W:M]\). Completing the crossed-product algebra at \([y]\) and splitting the points of \(Wy\) gives \begin{equation}\label{ord:local-matrix} \widehat{R\rtimes W}_{[y]} \cong\operatorname{Mat}_{n}(S_y\rtimes M). \end{equation} Choose \(r_z\in W\) with \(r_z z=y\) for each \(z\in Wy\). A finite affine parabolic, projected to \(H\subset W\), has a rational fixed point \(\xi\in X\otimes_{\ZZ}\QQ\); its elements are \(t_{\xi-h\xi}h\), with \(\xi-h\xi\in X\). Write \(u\) for the logarithmic coordinate at \(y\). The matrix entry from the \(z\)-factor to the \(hz\)-factor is \[ (hz)^{\xi-h\xi} \exp\bigl\langle r_{hz}\xi-mr_z\xi,u\bigr\rangle\,m, \qquad m=r_{hz}hr_z^{-1}\in M. \] Let \(g\) be the diagonal matrix with entries \(g_z=\exp\langle r_z\xi,u\rangle\). Conjugation by \(g^{-1}\) on the left and \(g\) on the right cancels the exponential, since \(m(g_z)=\exp\langle mr_z\xi,u\rangle\). The conjugated finite group therefore lies in \(\operatorname{Mat}_{n}(\CC[M])\), with the residual scalars \((hz)^{\xi-h\xi}\). Let \(e_F\in\CC[H]\) be the central idempotent of a full finite matrix block. In the semisimple algebra \(\operatorname{Mat}_{n}(\CC[M])\), the two-sided ideal generated by its image is the direct sum of the blocks on whose simple modules that image acts nontrivially. Indeed, a nonzero element of a simple matrix algebra generates that algebra as a two-sided ideal, and the central block projections separate the factors. Extending this equality to \(\operatorname{Mat}_{n}(S_y\rtimes M)\) shows that the full completed ideal is generated by the same constant blocks. Inner conjugation by \(g\) preserves every two-sided ideal. Proposition~\ref{ord:finite-generation} now proves constant-block generation for the completed ordinary based ideal. Let \(\mathfrak m_{S_y^M}\) be the maximal ideal of the invariant power-series ring. The central fiber of \eqref{ord:local-matrix} is \[ \operatorname{Mat}_{n}\bigl( (S_y/\mathfrak m_{S_y^M}S_y)\rtimes M\bigr). \] The positive-degree coinvariant ideal is nilpotent, and its quotient is \(\operatorname{Mat}_{n}(\CC[M])\). This quotient is semisimple, so the coinvariant ideal is the Jacobson radical. Every selected constant block therefore survives in the semisimple quotient of the central fiber. We next determine a necessary support condition for a selected block. Apply nonabelian completion localization to the Springer resolution and its Steinberg correspondence at \((y,1)\). Theorem~4.3(a) and Corollary~4.5(b) of \cite{EG} are natural for the proper map from the closed supported correspondence. They transport its image to the fixed supported correspondence over \(\cC_y\). The convolution normalization uses inverse normal Euler classes from the smooth fixed Springer resolution. These are invertible at \(y\) and preserve the supported image. Every component of the fixed flag variety is a flag variety for the connected reductive group \(G_y\) \cite[Proposition~8.8.7]{CG}. The fixed Springer object is thus a sum of \([W:M]\) copies of the ordinary Springer sheaf for \(G_y\), all with the same perverse shift. The total-extension description of the central Hecke fiber \cite[Proposition~8.1.5, Lemma~8.6.1 and Theorem~8.6.7]{CG} applies at \((y,1)\); see also \cite[Theorem~6.4 and its proof]{Antor}. It uses the simply connected derived group of the ambient \(G_{\CC}\). The maps through fixed-point \(K\)-theory and Borel--Moore homology commute with these proper supported maps and with restriction to an open subset of the base. Thus the image of the supported correspondence restricts to zero on \(\cN(\mathfrak g_y)\setminus\cC_y\). Write this semisimple perverse sheaf as \[ \mathcal S_y =\bigoplus_{(\cO,\mathscr L)} V_{\cO,\mathscr L} \otimes\IC(\overline{\cO},\mathscr L). \] On the complement of \(\cC_y\), exactly the summands with \(\cO\subseteq\cC_y\) vanish. The other summands remain nonzero and pairwise nonisomorphic simple perverse sheaves, because their dense orbits and local systems remain visible. Hence the kernel of restriction on degree-zero endomorphisms is \[ \bigoplus_{\cO\subseteq\cC_y,\mathscr L} \End(V_{\cO,\mathscr L}). \] Negative extensions vanish; the positive-degree ideal of the finite total Ext algebra is nilpotent. Its semisimple quotient is the degree-zero algebra. The central-fiber isomorphism preserves Jacobson radicals and induces an isomorphism of the two semisimple quotients. Consequently, a selected constant block must map into the displayed restriction kernel. This determines the allowed simple blocks. In the positive Hecke normalization a constant Hecke type \(F\) has the standard Springer type \(F\otimes\det_M\); the compatibility is recorded in \cite[Remark~4.1, Theorem~8.2 and \S12]{ABPS}. The antispherical right polynomial action is \(f\cdot w=\det(w)w^{-1}f\). These two twists cancel. The natural polynomial type \(E\) can therefore be selected only when \(\cO_M(E)\subseteq\cC_y\). If \(e_E\in\CC[M]\) is the central projector onto the natural polynomial type \(E\), then \[ (S_y\rtimes M)e_ES_y=S_yS_y[E]. \] Constant-block generation therefore gives \(\widehat I_{\cC,y}\subseteq K_{\cC,y}\). \end{proof} \subsection{Lowest Springer copies generate downsets} Let \(M\) now be any Weyl group, with reflection representation \(V\), and put \(S=\CC[V]\). For an orbit \(\cO\) in its ordinary nilpotent cone, let \(E_0(\cO)\) be the Springer representation for the trivial local system. Put \(d_{\cO}=\dim\mathcal B_{\cO}\). The proof below shows that \(E_0(\cO)\) first occurs once in polynomial degree \(d_{\cO}\). Denote that copy by \(U_{\cO}\). \begin{proposition}\label{ord:kato} For every nilpotent downset \(\mathscr D\), \begin{equation}\label{ord:kato-formula} \sum_{\cO_M(E)\subseteq\mathscr D} S\,S[E] =\sum_{\cO\subseteq\mathscr D}S\,U_{\cO}. \end{equation} This also holds after adding variables on which \(M\) acts trivially and after completion at the origin. \end{proposition} \begin{proof} Consider \(B=S\rtimes\CC[M]\). The natural polynomial module \(S\) is the projective \(B\)-module with trivial head. The Springer extension-algebra description and canonical trace filtration are given by \cite[Theorems~3.1 and~3.5]{KatoGreen} and \cite[Theorem~1.8, Remark~1.9, Corollary~3.12 and Theorem~4.1]{KatoExt}. These results apply to the ordinary Springer block and its orbit-closure order. Choose a total order \(\cO_1,\ldots,\cO_m\) refining closure, with all orbits of \(\mathscr D\) first. Let \(e_j\in\CC[M]\) be the sum of the central matrix-block idempotents for the Springer pairs on \(\cO_j\), and set \[ F_0=0,\qquad F_j=B(e_1+\cdots+e_j)S. \] Since each central projector extracts its full isotypic component, \(F_j\) is exactly the sum of \(S\,S[E]\) over these first orbits. The proof of \cite[Theorem~4.1]{KatoExt} constructs the quotients of a projective by its earlier trace submodules. Applied to \(S\), it identifies \(F_j/F_{j-1}\) with a direct sum of shifted modules \(\widetilde K_{\cO_j,\mathscr L}\). Explicitly this is the actual kernel \[ \ker\bigl(S/F_{j-1}\longrightarrow S/F_j\bigr) =F_j/F_{j-1} \] of the consecutive projective trace-quotient maps. Using full central projectors gives the same trace submodule as using one primitive projector for each simple type, since the corresponding matrix units generate one another under multiplication by \(B\). Each \(\widetilde K_{\cO_j,\mathscr L}\) is generated in degree zero by its irreducible Springer head and is a quotient of the corresponding graded projective. It is concentrated in nonnegative degrees and is annihilated by \(e_1,\ldots,e_{j-1}\). The multiplicities in the trace quotient are given by Brauer--Humphreys reciprocity. The trivial Weyl representation occurs in Springer cohomology exactly once, in \(H^0(\mathcal B_x)\), and the component group acts trivially on that line \cite[\S\S2.3 and~2.9]{BM}. Reciprocity therefore gives \begin{equation}\label{ord:trace-quotient} F_j/F_{j-1}\cong \widetilde K_{\cO_j,1}\langle d_{\cO_j}\rangle, \end{equation} where the shift places the head in polynomial degree \(d_{\cO_j}\). In particular, that head is a single copy of \(E_0(\cO_j)\). To locate its lift inside \(S\), fix \(\cO\) and choose the order with \(\overline{\cO}\) first. Every orbit strictly below \(\cO\) has larger Springer-fiber dimension. Its shifted trace standard therefore contributes no vector in degrees \(\leq d_{\cO}\). Every later trace standard is annihilated by the earlier projector onto \(E_0(\cO)\). Exactness of this projector on each graded piece and \eqref{ord:trace-quotient} show that \(E_0(\cO)\) occurs in \(S\) with multiplicity zero below degree \(d_{\cO}\) and with multiplicity one in that degree. This defines \(U_{\cO}\). Return to the order with \(\mathscr D\) first. Semisimplicity of \(\CC[M]\) lifts the head in \eqref{ord:trace-quotient} to \(U_{\cO_j}\subset F_j\). Since that quotient is generated by its head, \[ F_j=F_{j-1}+S\,U_{\cO_j}. \] Induction through the orbits of \(\mathscr D\) proves \eqref{ord:kato-formula}. Polynomial extension and completion are flat and preserve the finite generating sum. \end{proof} \begin{lemma}\label{ord:nakayama} Let \(S=\CC[[V]]\), with maximal ideal \(\mathfrak m\), and let \(K\) be the completion of the ideal in \eqref{ord:kato-formula}. Suppose \(J\subseteq K\) is an \(M\)-stable ideal. If, for every surviving type \(U_{\cO}\) in \(K/\mathfrak mK\), an element of \(J\) has nonzero degree-\(d_{\cO}\) term in that type, then \(J=K\). \end{lemma} \begin{proof} There is a graded equivariant surjection \[ \bigoplus_{\cO}U_{\cO} \longrightarrow K/\mathfrak mK . \] The types \(E_0(\cO)\) are pairwise nonisomorphic. A surviving type therefore occurs just once, in its single displayed degree. Project the given element of \(J\) to that type by a finite group average. Its higher terms are in \(\mathfrak mK\). The infinite tail is also there, because this submodule is adically closed. Its \(M\)-span fills the indicated summand of \(K/\mathfrak mK\). Thus \(J+\mathfrak mK=K\); Nakayama applied to the finite module \(K/J\) proves the claim. \end{proof} For actual source families with quantum support in \(\cC\), let \(J_{\mathrm{source},y}\) be the sum of their completed coefficient ideals. Then \begin{equation}\label{ord:sandwich} J_{\mathrm{source},y} \subseteq\widehat I^+_{\cC,y} \subseteq\widehat I_{\cC,y} \subseteq K_{\cC,y}. \end{equation} The first two inclusions are supplied by actual source support; the last is Proposition~\ref{ord:containment}. It is therefore enough to obtain a nonzero degree-\(d_{\cO}\) component in every surviving generator type \(U_{\cO}\) of \(K_{\cC,y}/\mathfrak m_yK_{\cC,y}\). % End sections/ordinary.tex % Begin sections/local_global.tex \section{Finite sources, retaining cuts, and globalization} \label{sec:local-global} \subsection{From a finite parity object to an actual source} Let \(H\) be a finite affine parabolic, and suppose a finite canonical element \(C_x\) is the character of an indecomposable parity object over the modular field. Thus \({}^pC_x=C_x\). Tensor this object with arbitrary Bott--Samelson prefixes and suffixes, and pass to the antispherical category. The resulting characters have nonnegative Laurent coefficients in the \(p\)-canonical basis. At \(v=1\) they are therefore characters of actual tilting modules \cite[Lemma~5.8 and Theorem~7.2]{AHR}. Their ordinary characters belong to the based two-sided ideal generated by \(C_x\). Hence every indecomposable summand of their quantum lifts has support in the closure of the ordinary cell of \(x\). Theorem~\ref{lift:whole} gives the same bound for these modular tilting modules and their direct summands. The character action of the fixed parity object lies in the linear span of Bott--Samelson character actions, since \(C_s(1)=1+s\). At \(v=1\), the two-sided ideal of \(\CC[H]\) generated by \(C_x(1)\) contains exactly the complete matrix blocks on which \(C_x(1)\) acts nontrivially. Every finite group element is a product of \(s=C_s(1)-1\). Hence those matrix blocks are in the complex span of the stated actual prefixes and suffixes, with the same support bound. Left multiplication is performed in the regular Hecke category before passing to the antispherical module. Thus the complete finite blocks belong to the same bounded character span. Its closure under tensor and linkage character operators still has this support bound, so Proposition~\ref{op:coefficient} applies to that stable span. An \emph{actual finite block source} will mean a block obtained in this manner, with its proven ordinary support bound. More than one \(p\)-independent canonical vector may be used to obtain it. The exact-family selection in the classical construction is an instance of this definition. \subsection{The finite trace formula} Project the finite affine parabolic to \(H\subset W\), and choose its rational fixed point \(\xi\). Its elements have the form \(t_{\xi-h\xi}h\), where \(\xi-h\xi\in X\). Let \(y\in\mathbb T\) have stabilizer \(M\). For each \(z\in Wy\), choose \(r_z\in W\) with \(r_z z=y\). For \(E\in\Irr(M)\), define \begin{equation}\label{loc:trace} \chi_{E,y,\xi}(h)= \sum_{\substack{z\in Wy\\hz=z}} z^{\xi-h\xi}\, \chi_E(r_zhr_z^{-1}). \end{equation} \begin{lemma}\label{loc:compression} In the standard Springer normalization, a full \(H\)-matrix block \(F\) acts nontrivially on the simple constant matrix module labelled by \(E\) in \eqref{ord:local-matrix} if and only if \begin{equation}\label{loc:multiplicity} m(F,E;y,\xi)= \frac1{|H|}\sum_{h\in H} \overline{\chi_F(h)}\,\chi_{E,y,\xi}(h)>0. \end{equation} When this condition holds, the coefficient ideal of the source contains the entire completed natural isotypic ideal \(S_yS_y[E]\). \end{lemma} \begin{proof} After the determinant twists fixed in Proposition~\ref{ord:containment} cancel, the constant matrix module is a direct sum of copies of \(E\), indexed by \(Wy\). The affine element \(t_{\xi-h\xi}h\) sends the copy at \(z\) to the copy at \(hz\); its diagonal entry when \(hz=z\) is \[ z^{\xi-h\xi}\,r_zhr_z^{-1}. \] Its trace is therefore \eqref{loc:trace}. Changing \(r_z\) conjugates the corresponding \(M\)-action and preserves this trace. The character inner product \eqref{loc:multiplicity} is the multiplicity of \(F\) in this finite-dimensional \(H\)-module. Hence the \(F\)-block acts nontrivially exactly when that multiplicity is positive. This is Mackey restriction through the full subgroup \(H\cap r_z^{-1}Mr_z\), with the displayed scalar character. The constant matrix ideal then contains the \(E\)-block by semisimplicity. The matrix argument in Proposition~\ref{ord:containment} extends that inclusion to the completed algebra, and its action on the natural polynomial module gives \(S_yS_y[E]\). \end{proof} At \(y=1\), formula~\eqref{loc:multiplicity} reduces to the multiplicity of \(E\) in \(\Ind_H^W F\). This is also a normalization check on the determinant twist. The positive Hecke block used as an actual object is \(\det_H\otimes F\), while the formula itself uses standard Springer labels. \subsection{Native discriminant sources} An ordinary parabolic \(L\) of \(M\) may occur as the stabilizer of a \(p\)-th root of \(y\). The longest element of its corresponding finite affine stabilizer has a smooth parity object. Its standard Springer block is \(\det_L\). If that affine parabolic is not standard, start with the conjugate standard longest source and apply the conjugating group elements on its two sides in the character span. Writing \(s=C_s(1)-1\) expresses these as finite combinations of actual Bott--Samelson operations, preserving the ordinary two-sided support bound. For the simply laced exceptional full-rank points, the residual scalar equals \(1\). Normalize the invariant form so that every root has squared length \(2\), and use it to identify the weight and cocharacter spaces in this paragraph. A root sheet is torsion, so write \(t=\exp(2\pi i\beta)\) with \(\beta\in Y\otimes_{\ZZ}\mathbb R\). Put \(Q^\vee=\sum_{\alpha\in\Phi}\ZZ\alpha^\vee\). Since the ambient group is simply connected, \(Y=Q^\vee\); thus \(\beta-h\beta\in Q^\vee=Q\) for its stabilizer. Since \(h\) is orthogonal, \[ \exp\bigl(2\pi i(\beta,\beta-h\beta)\bigr) =\exp\bigl(\pi i\|\beta-h\beta\|^2\bigr)=1, \] because the simply laced root lattice is even. The same equality holds at \(y=t^p\). After the exponential gauge is removed, the local coefficient ideal therefore contains \(\Delta_L\). Its support bound is \[ \mathcal C_L =\overline{\cO(j_L^W\det_L)} \] by Proposition~\ref{ord:finite-label}. Corollary~\ref{op:retention} permits this bound to cut an existing source while retaining every \(E\)-type with \(\Hom_L(\det_L,\Res^M_L E)\ne0\). Here is the root-sheet existence argument for the exceptional full-rank points used below. Their orders are among \(1,2,3,4,5,6\), hence prime to \(p\). Choose a \(p\)-th root \(y_0\) that is a power of \(y\), so \(M\) fixes \(y_0\). For a standard parabolic \(L\subset M\), prescribe root pairings zero on its simple roots and one on the other simple roots of \(M\), modulo \(p\). The relevant Cartan and lattice indices have prime divisors at most \(7\); they are invertible in the \(E_7,E_8\) ranges \(p>18,p>30\). There is consequently a \(p\)-torsion point \(z\) with these pairings. For a positive root of \(M\) outside \(L\), the sum of the prescribed simple coefficients is between \(1\) and \(h(M)-10,\qquad \Hom_{L_j}(\det_{L_j},E_0(\cO))\ne0, \] and \begin{equation}\label{loc:cut-bound} \mathcal C_F\cap\bigcap_{j=1}^r\mathcal C_{L_j} \subseteq \overline{G_{\CC}\cdot\cO}. \end{equation} Then \(\widehat I^+_{\cC,y}=\widehat I_{\cC,y}\) for every downset \(\cC\). It is also enough to construct the nonzero minimum jets in Lemma~\ref{ord:nakayama} by actual finite tensor source words with these bounds. \end{proposition} \begin{proof} Lemma~\ref{loc:compression} supplies the entire \(E_0(\cO)\)-isotypic ideal from \(F\). Each cut retains it by Corollary~\ref{op:retention}, while intersecting the source bounds. Thus every minimum generator for the local downset \(\cC_y\) lies in \(\widehat I^+_{\cC,y}\). Use Proposition~\ref{ord:kato} and the containment \eqref{ord:sandwich}. For the minimum-jet formulation, apply Lemma~\ref{ord:nakayama} to the same sandwich. \end{proof} \subsection{Central translation} \begin{lemma}\label{loc:center} The coefficient ideals of actual finite source words, at their specified support bounds, are stable under the finite center \(Z=\Hom(X/Q,\CC^\times)\). The same is true of the ordinary cell-downset coefficient ideals. \end{lemma} \begin{proof} Take the actual-operator closure of each finite source word at its proved support bound. Tensoring and linkage projection preserve that bound, so its character space is an \(E_0\)-submodule. Proposition~\ref{op:coefficient} makes its coefficient ideal invariant under \(Z\). The ordinary cell-downset ideal corresponds to the operator-stable quantum support space \(J^q_{\cC}\), so the same proposition applies. \end{proof} \subsection{Constancy and the finite full-rank test} Stratify \(\mathbb T\) by the connected components of intersections of root hypertori \(\alpha(y)=1\), with all further root hypertori removed. This is a finite stratification. On a stratum the subgroup \(M=W_y\) is constant. For a fixed finite affine source, and \(h\) in a fixed intersection with \(M\), the function \[ y\longmapsto y^{\xi-h\xi} \] is a regular Laurent monomial. If \(h\) has order \(r\), the product of its \(r\) translates telescopes to \(1\). Since \(h\) fixes \(y\), this function is an \(r\)-th root of unity. It is therefore constant on the connected stratum. The full finite trace formula and its exponential gauge give one fixed completed ideal in logarithmic coordinates throughout that stratum. This property is preserved by finite sums and finite tensor words. To see the tensor step, pull back to the finite etale \(p\)-power cover. For a root of \(M\), its value on a root sheet is in \(\mu_p\), hence is locally constant. A root outside \(M\) cannot fix a sheet. Thus the multiset of the groups \(W_t\) is locally constant; monodromy can only permute the sheets. Formula \eqref{op:tensor-formula} sums over all of them and is consequently constant as an ideal in the common logarithmic coordinate ring. The sum includes every root-sheet stabilizer. The retaining cuts use the ordinary parabolics specified in their construction. \begin{lemma}\label{loc:boundary} The closure of each root-stabilizer stratum contains a point whose root stabilizer has full rank. \end{lemma} \begin{proof} Write the closure as a connected subtorus coset \(a\mathbb T_0\), and put \(\ell=\operatorname{rank}X\). Choose independent roots \(\alpha_1,\ldots,\alpha_r\) that equal \(1\) on this coset and span its normal space. Extend them to a full-rank set \(\alpha_1,\ldots,\alpha_\ell\). The restrictions of \(\alpha_{r+1},\ldots,\alpha_\ell\) to \(\mathbb T_0\) are independent and span its rational character space. They define an isogeny \[ \mathbb T_0\longrightarrow(\CC^\times)^{\ell-r}, \qquad t\longmapsto \bigl(\alpha_{r+1}(t),\ldots,\alpha_\ell(t)\bigr). \] Choose \(t\) mapping to \(\bigl(\alpha_{r+1}(a)^{-1},\ldots,\alpha_\ell(a)^{-1}\bigr)\). The point \(at\) lies on all \(\ell\) root hypertori, so its root stabilizer has full rank. \end{proof} \begin{theorem}\label{loc:globalization} Suppose finite actual source words satisfy Proposition~\ref{loc:finite-criterion} at every full-rank point, up to Weyl conjugacy and central translation. Then \[ I^+_{\cC}=I_{\cC} \] for every downset \(\cC\). \end{theorem} \begin{proof} A full-rank point satisfies a full-rank set of root equations, so it is torsion. Write it as \(y=\exp(2\pi i\beta)\), with \(\beta\in Y\otimes_{\ZZ}\mathbb R\). Here the exponential has kernel \(Y=Q^\vee\). Consequently Weyl-conjugacy classes are represented in a closed alcove for \(Q^\vee\rtimes W\) acting on the cocharacter space \(Y\otimes_{\ZZ}\mathbb R\). A full-rank root stabilizer fixes a zero-dimensional face, hence a vertex. There are therefore finitely many such points, and Lemma~\ref{loc:center} permits central translation among their representatives. For a fixed \(\cC\), take the finite sum \(L_{\cC}\) of the source-word ideals providing all required generators at these points. It has the established bound, so \(L_{\cC}\subseteq I^+_{\cC} \subseteq I_{\cC}\). The ordinary ideal has a constant formal expression on each stratum by Proposition~\ref{ord:finite-generation} and finite block compression. The preceding constancy argument applies to \(L_{\cC}\) as well. The support of the finite module \(I_{\cC}/L_{\cC}\) is closed. On any connected stratum it is either empty or the whole stratum, because completed inclusion is constant and completion is faithfully flat. If it were nonempty, its closure would contain a full-rank point by Lemma~\ref{loc:boundary}. This contradicts the verified equality there. Thus \(L_{\cC}=I_{\cC}\), which proves the theorem. \end{proof} % End sections/local_global.tex % Begin sections/classical.tex \section{Positive sources in the classical types}\label{cl:section} We construct the local minimum Springer copies required in Proposition~\ref{ord:kato}. All coefficient ideals and polynomial representations in this section are over \(\mathbb C\). Source objects are parity objects with coefficients of characteristic \(p\), and give tilting objects under the categorical realization. For a finite reflection group \(K\), write \(\varepsilon_K\) for its determinant character and \(\Delta_K\) for the product of its positive roots. For a finite reflection subgroup \(L\), set \[ \operatorname{Alt}_L(f)=\frac1{|L|}\sum_{w\in L}\det(w)w(f). \] We use the standard Springer convention, in which the regular nilpotent orbit corresponds to the trivial representation. Thus, if \(E\) is a representation on which a positive Hecke element \(C_s(1)=1+s\) acts, its standard Springer label is \(E\otimes\varepsilon_K\). We call the latter the \emph{standard block} of the source. For disjoint sets of variables \(I\) and \(J\), put \begin{equation}\label{cl:column-polynomials} D_a(I)=\prod_{\substack{i|\beta|\); the two type-\(B\) extensions have minimum degrees \(b(\alpha,\beta)\) and \(b(\beta,\alpha)\), with \(b(\beta,\alpha)-b(\alpha,\beta)=|\alpha|-|\beta|>0\). Restriction of the lower-degree polynomial span therefore gives the unique minimum copy. A label with \(\alpha=\beta\) is called \emph{balanced}, or \emph{degenerate}; its restriction has two irreducible halves. Their minimum copies are treated separately below. \subsection{From finite parity objects to complete matrix blocks} Write \(\mathcal H_{\mathrm{ext}}\) for the extended Hecke algebra. At \(v=1\), its antispherical quotient map is \[ \pi:\mathcal H_{\mathrm{ext}}(1)\longrightarrow \operatorname{sgn}\otimes_{\mathbb C[W]} \mathbb C[W_{\mathrm{ext}}], \qquad \pi(h)=1\otimes h. \] We also write \(\pi\) for the corresponding quotient functor on Hecke objects. \begin{lemma}\label{cl:matrix-block} Let \(K\) be a finite standard affine parabolic subgroup, and let \(Z\) be an actual object of its finite Hecke category, with character \(z\in\mathcal H_K\). Suppose that \(z\) belongs to an ordinary based two-sided ideal \(\mathcal H_{\mathcal U}\) of the ambient Hecke algebra and that \(z(1)\) acts nontrivially on an irreducible \(\mathbb C[K]\)-module \(E\). The character span of actual sources with ordinary support in \(\mathcal U\) contains the entire antispherical image \[ \pi\bigl(\mathcal H_{\mathrm{ext}}(1)e_E \mathcal H_{\mathrm{ext}}(1)\bigr), \] where \(e_E\) is the central block idempotent of \(E\). Consequently it contains the full finite-block coefficient ideal, with all its completed local degrees. \end{lemma} \begin{proof} The nonzero matrix \(z(1)|_E\) generates \(\operatorname{End}_{\mathbb C}(E)\) as a two-sided ideal. Semisimplicity of \(\mathbb C[K]\) therefore gives \[ e_E\in\mathbb C[K]z(1)\mathbb C[K]. \] The characters of Bott--Samelson objects span the Hecke algebra: the character of a reduced Bott--Samelson word has its corresponding standard basis element with coefficient \(1\), and all other terms have smaller length. Include the invertible length-zero objects to obtain the extended algebra. Hence every element of the displayed antispherical image is a finite complex linear combination of characters of objects \[ \pi\bigl(B_{\underline x}\otimes Z\otimes B_{\underline y}\bigr). \] Here the left factor is applied in the regular category before taking the antispherical quotient. The quotient sends an indecomposable to its antispherical indecomposable or to zero. Because \(\mathcal H_{\mathcal U}\) is a based two-sided ideal, all ordinary character terms in these products lie in \(\mathcal U\). The expansions of actual objects in the modular basis, and of the modular basis in the ordinary basis, have nonnegative coefficients. Thus no actual summand can acquire an ordinary term outside \(\mathcal U\). The coefficient correspondence of Proposition~\ref{op:coefficient} now gives the asserted ideal inclusion. The idempotent calculation thus gives complex linear relations among characters of actual objects. \end{proof} The finite affine-parabolic label rule will be used throughout this section. If \(F=E\otimes\varepsilon_K\) and \(F_{\mathrm{sp}}\) is the special representation in its ordinary family, the ambient ordinary cell of a source in that family is the Springer orbit of \begin{equation}\label{cl:finite-label-rule} j_K^W(F_{\mathrm{sp}}). \end{equation} Here \(j\) denotes truncated induction in minimum polynomial degree. This is the finite-parabolic cell theorem of \cite[\S\S5.4--5.6, 6.5 and Lemma 7.2]{LusztigIV}. For standard Levi factors, truncated induction is compatible with ordinary induction of nilpotent orbits. In the classical partition notation, a symmetric-group longest factor on \(m\) coordinates appends the two columns \((m,m)\), followed by the appropriate classical collapse. We will compute the special family before applying \eqref{cl:finite-label-rule}. \begin{proposition}\label{cl:minuscule} At every odd coefficient characteristic the following standard blocks have actual finite sources in their exact ordinary families: \[ ((1^a),(1^b))\quad\text{in }W(B_{a+b}),\qquad \{(1^a),(1^b)\}\quad\text{in }W(D_{a+b}). \] For a balanced type-\(D\) block both halves are available, using the two fork parabolics. Arbitrary symmetric-group longest factors may be adjoined, and the complete corresponding matrix-block ideal is then available. \end{proposition} \begin{proof} Let \(A=\mathfrak S_s\) be the permutation parabolic of \(B_s\), or one of the two spinor parabolics of \(D_s\). The spherical parity theorem for minuscule flag varieties gives the ordinary spherical canonical basis at every good prime, hence at every odd prime in these types \cite[Theorem 9.1]{Baine}. The normalized smooth pullback to the full flag variety sends the spherical basis vector indexed by \(u\) to \(C_{uw_A}\). Its character formula is independent of the coefficient field. In characteristic zero it is one IC character; nonnegativity of the modular-to-ordinary transition then implies that the modular pullback is the same single canonical character. Consequently all the full-flag canonical vectors in this spherical module are actual \(p\)-independent sources. Fourier decomposition for the normal sign subgroup gives \[ \operatorname{Ind}_{\mathfrak S_s}^{B_s}\mathbf1 =\bigoplus_{b=0}^s E_{((s-b),(b))}. \] Restriction to \(D_s\) identifies complementary nonbalanced labels. For \(s=2b\), the two fork embeddings supply the two balanced halves. After tensoring by the full determinant these are precisely the standard blocks in the statement. It remains to select a source in the exact family of the desired block. The spherical module \(\mathcal H C_{w_A}\) and an ordinary based two-sided ideal are coordinate subspaces of the same canonical basis. Their intersection therefore has as a basis exactly the canonical vectors belonging to both. Intersect with the downset of the family of \(E\). Its exact family quotient has a nonzero \(E\)-row, since \(E^A\neq0\). At least one canonical vector in that quotient acts nontrivially on \(E\). Vectors in strictly lower families annihilate \(E\), so the chosen vector belongs to the desired family itself. Applying Lemma~\ref{cl:matrix-block} gives the complete block. External products with smooth symmetric-group longest objects preserve all these conclusions. \end{proof} \begin{remark}\label{cl:rank-one} We use \(D_2=A_1\times A_1\) and \(D_3=A_3\) with their usual interpretations. A displayed \(D_1\) factor denotes one unused coordinate with the identity object. The identity object also handles an empty block. \end{remark} \subsection{Orthogonal rank varieties and their integral forms} The remaining orthogonal mixed pairs use pure two-row blocks. We construct them by computing every graded local intersection form. Set \(\Lambda=\mathbb Z[1/2]\). Let \(A\geq R\geq1\), let \(n=A+R\), and fix an isotropic \(R\)-plane \(Z\) in a quadratic space of dimension \(2n+1\). Choose a Witt splitting \[ Z^\perp=Z\oplus V,\qquad \dim V=2A+1. \] Inside \(\operatorname{OG}(R,Z^\perp)\), define \begin{equation}\label{cl:rank-chain} X_j=\{U:\dim(U\cap Z)\geq j\},\qquad 0\leq j\leq R. \end{equation} Set \(X_{R+1}=\varnothing\). These are Schubert varieties for an isotropic flag containing \(Z\). The stratum \(S_k=X_k\setminus X_{k+1}\) is an affine bundle over \[ \operatorname{Gr}(k,R)\times\operatorname{OG}(R-k,2A+1). \] Indeed \(U\cap Z\) and the image of \(U\) in \(V\) determine the base point, and the remaining datum is a linear graph map. In particular the strata are simply connected. If \(c=A-R\), then \begin{equation}\label{cl:rank-codimension} \dim X_0=2AR-\frac{R(R-1)}2,\qquad \operatorname{codim}_{X_0}S_k =2ck+\frac{k(3k+1)}2. \end{equation} For \(0\leq r\leq R\), let \(\mathcal U_r\) denote the tautological bundle on \(\operatorname{OG}(r,V)\). The incidence morphism \begin{equation}\label{cl:rank-resolution} Y_r=\operatorname{Gr}_{\operatorname{OG}(r,V)} (R,Z\oplus\mathcal U_r) \longrightarrow X_{R-r},\qquad (U,L)\longmapsto U \end{equation} has smooth projective source and is birational onto its image. Over a point whose projection to \(V\) has rank \(j\), its fiber is \(\operatorname{OG}(r-j,2(A-j)+1)\). \begin{lemma}\label{cl:resolved-slice} Transverse to the projection-rank-\(j\) stratum, the resolved slice of \eqref{cl:rank-resolution} is \[ \operatorname{Tot}\operatorname{Hom} \bigl(\mathbb C^{R-j},\mathcal U_{r-j}\bigr) \longrightarrow \{B:\mathbb C^{R-j}\longrightarrow \mathbb C^{2(A-j)+1}:B^{\mathsf t}QB=0,\ \operatorname{rank}B\leq r-j\}. \] Here \(Q\) is a Gram matrix of the residual quadratic space. The fiber over the origin is the zero section. \end{lemma} \begin{proof} Choose a basis of the rank-\(j\) isotropic image and extend it to a Witt decomposition. On the open chart in which the corresponding rank-\(j\) minor is invertible, row and column elimination normalize that minor to the identity. The Gram equations involving one of these \(j\) columns solve uniquely for its coordinates in the dual isotropic summand. These solved coordinates and the remaining graph coordinates are smooth parameters. The unsolved columns form a map from \(\mathbb C^{R-j}\) to the nondegenerate residual space of dimension \(2(A-j)+1\), and their only remaining equations are \(B^{\mathsf t}QB=0\). In the incidence resolution, the isotropic \(r\)-plane containing the image descends, after removing the normalized \(j\)-plane, to an isotropic \((r-j)\)-plane containing the residual image. Over the Grassmannian of such planes the residual map is unrestricted. Thus the resolved slice is the displayed vector bundle. \end{proof} Write \(k=r-j\), \(a=A-j\), and \(K=R-j\). The top Chern class of \(\operatorname{Hom}(\mathbb C^K,\mathcal U_k)\) is \(z^K\), where \(z=c_k(\mathcal U_k)\). The self-intersection formula for the zero section, followed by integral Poincare duality on the smooth Grassmannian, identifies every graded local form with a graded piece of \begin{equation}\label{cl:euler-map} z^K:H^*(\operatorname{OG}(k,2a+1);\Lambda) \longrightarrow H^*(\operatorname{OG}(k,2a+1);\Lambda). \end{equation} Poincare duality is unimodular over \(\Lambda\). A submodule over \(\Lambda\) is \emph{saturated} if its quotient is torsion-free. \begin{lemma}\label{cl:odd-euler} For every \(a\geq k\geq1\) and every \(K\geq0\), the image of every graded piece of \eqref{cl:euler-map} is saturated over \(\Lambda\). \end{lemma} \begin{proof} Let \(x_1,\ldots,x_k\) be the Chern roots of \(\mathcal U_k\), and put \(c=a-k\). Write \(e_i\) and \(h_i\) for the elementary and complete symmetric functions, defined by \[ \sum_{i\geq0}e_i(x^2)u^i=\prod_{j=1}^k(1+x_j^2u),\qquad \sum_{i\geq0}h_i(x^2)u^i=\prod_{j=1}^k(1-x_j^2u)^{-1}. \] Thus \(e_0=h_0=1\); negative-index complete symmetric functions are zero. The Borel presentation, after inverting the torsion index, is \begin{equation}\label{cl:odd-borel} H^*(\operatorname{OG}(k,2a+1);\Lambda) =\frac{\Lambda[x_1,\ldots,x_k]^{\mathfrak S_k}} {(h_{c+1}(x^2),\ldots,h_{c+k}(x^2))}. \end{equation} Only the prime \(2\) divides the relevant orthogonal torsion indices. For this paragraph let \(G_0\) be a simply connected complex group of type \(B_m\) or \(D_m\). Choose a maximal torus \(T_0\subset B_0\subset P_0\), where \(B_0\) is a Borel subgroup and \(P_0\) is a parabolic subgroup of \(G_0\). Let \(W_0\) be the Weyl group of \(G_0\), and \(W_P\) that of the Levi subgroup of \(P_0\) containing \(T_0\). After tensoring with \(\Lambda\), identify \(X^*(T_0)\) with the orthogonal coordinate lattice, and put \(\mathscr P=\Lambda[x_1,\ldots,x_m]\). A subscript \(+\) denotes the positive-degree ideal. Write \(\operatorname{cl}:\mathscr P\to H^*(G_0/B_0;\Lambda)\) for the characteristic map and \(\partial_s(f)=(f-sf)/\alpha_s\) for the divided difference, where \(\alpha_s\) is the corresponding simple root. The additive graded form of \cite[Theorem 6.7, Remark 6.8, and Corollaries 13.9--13.10]{CPZ}, together with the Schubert-cell identification of Chow and singular cohomology over \(\mathbb C\), gives \[ H^*(G_0/B_0;\Lambda) \cong\mathscr P/(\mathscr P^{W_0}_{+})\mathscr P \] and makes \(\mathscr P\) finite free over \(\mathscr P^{W_0}\). Every torsion index used here, including that of the semisimple part of the Levi, is a power of \(2\). Write \(\partial_{w_P}\) for the longest divided-difference operator. The characteristic map for the Levi flag variety supplies a homogeneous polynomial \(q_P\) with \(\partial_{w_P}(q_P)=1\): an integral representative whose pushforward is the Levi torsion index can be divided by that index in \(\Lambda\). Consequently \[ \Pi_P(f)=\partial_{w_P}(q_Pf) \] is a \(\mathscr P^{W_0}\)-linear projection from \(\mathscr P\) onto \(\mathscr P^{W_P}\). Indeed \(\partial_{w_P}\) takes values in the \(W_P\)-invariants, and for invariant \(f\) the expression is \(f\partial_{w_P}(q_P)=f\). If \(\pi:G_0/B_0\longrightarrow G_0/P_0\) is the flag projection, the geometric divided-difference identity identifies this projection in cohomology with \[ a\longmapsto\pi^*\pi_*(\operatorname{cl}(q_P)a). \] The Schubert classes of the Levi flag variety lift to \(G_0/B_0\) after inverting its torsion index. Leray--Hirsch therefore makes \(\pi^*\) injective. The displayed operator has image exactly \(\pi^*H^*(G_0/P_0;\Lambda)\), since \(\pi_*\operatorname{cl}(q_P)=1\). Its polynomial version, being \(\mathscr P^{W_0}\)-linear, also gives \[ \mathscr P^{W_P}\cap (\mathscr P^{W_0}_{+})\mathscr P =(\mathscr P^{W_0}_{+})\mathscr P^{W_P}. \] It follows that the partial-flag characteristic presentation is \[ H^*(G_0/P_0;\Lambda) \cong \mathscr P^{W_P}/ (\mathscr P^{W_0}_{+})\mathscr P^{W_P}. \] For \(W_0=B_a\) and \(W_P=\mathfrak S_k\times B_{a-k}\), elimination of the residual squared-root elementary functions gives \eqref{cl:odd-borel}. For \(W_0=D_a\) and \(W_P=\mathfrak S_k\times D_{a-k}\), the same elimination retains the residual Euler class; the full Euler relation is \(zw=0\), and the remaining relations are exactly those of \eqref{cl:even-borel}. The case \(a-k=1\) has a residual torus and is included by taking its coordinate as \(w\). Apply the same argument with \(W_0=D_k\) and \(W_P=\mathfrak S_k\), for \(k\geq2\). Put \[ \mathscr P_k=\Lambda[x_1,\ldots,x_k],\qquad \mathscr P_k^{D_k}=\Lambda[e_1(x^2),\ldots,e_{k-1}(x^2),z]. \] In this case one may take \(q_P=x_1^{k-1}x_2^{k-2}\cdots x_{k-1}\), with the sign chosen according to the positive-root convention, so no division by \(k!\) is involved. The projection \(\Pi_P\) makes \(\mathscr P_k^{\mathfrak S_k}\) a direct summand of the finite free \(\mathscr P_k^{D_k}\)-module \(\mathscr P_k\). Thus \(\mathscr P_k^{\mathfrak S_k}\) is projective over \(\mathscr P_k^{D_k}\). Its quotient by \((\mathscr P_k^{D_k})_+\) is the cohomology of one spinor partial flag variety, which has a homogeneous Schubert basis of size \[ [D_k:\mathfrak S_k]=2^{k-1}. \] Lift this basis homogeneously to \(\mathscr P_k^{\mathfrak S_k}\). Graded Nakayama gives a surjection from the free \(\mathscr P_k^{D_k}\)-module on these lifts to \(\mathscr P_k^{\mathfrak S_k}\). Projectivity makes its kernel a direct summand; reducing the split sequence modulo \((\mathscr P_k^{D_k})_+\) shows that this kernel has zero quotient. Graded Nakayama applied again makes the kernel zero. This proves the asserted homogeneous freeness over \(\mathscr P_k^{D_k}\), with rank \(2^{k-1}\). For \(k=2\) the same proof uses \(D_2=A_1\times A_1\). Set \[ B=\frac{\Lambda[e_1(x^2),\ldots,e_k(x^2)]} {(h_{c+1}(x^2),\ldots,h_{c+k}(x^2))} \cong H^*(\operatorname{Gr}(k,a);\Lambda), \] where the grading on the last ring is doubled. The invariant tower \(\mathfrak S_k\subset D_k\subset B_k\) gives \[ \Lambda[x]^{D_k} =\Lambda[x]^{B_k}[z]/(z^2-e_k(x^2)), \qquad z=x_1\cdots x_k. \] By the freeness just proved, \eqref{cl:odd-borel}, as a module over \(B[z]/(z^2-e_k)\), is a direct sum of homogeneous shifts of that ring. For \(k=1\) this assertion is simply \(\Lambda[x]=\Lambda[x^2]\oplus x\Lambda[x^2]\); for \(k=2\) it uses \(D_2=A_1\times A_1\). If \(K=2m\), multiplication by \(z^K\) is \(e_k^m\) on each of the two summands \(B\) and \(zB\). If \(K=2m+1\), its matrix is \begin{equation}\label{cl:odd-euler-matrix} \begin{pmatrix}0&e_k^{m+1}\\ e_k^m&0\end{pmatrix}. \end{equation} In the integral Schur basis of the \(k\times c\) Grassmannian rectangle, \begin{equation}\label{cl:rectangle-shift} e_k^m s_\lambda= \begin{cases} s_{\lambda+(m^k)},&\lambda+(m^k)\subseteq(c^k),\\ 0,&\text{otherwise}. \end{cases} \end{equation} Here \(s_\lambda\) is the Schur polynomial in \(x_1^2,\ldots,x_k^2\). Multiplication by \(e_k(x^2)^m=(x_1\cdots x_k)^{2m}\) adds \(m\) to every part of \(\lambda\); the Grassmannian relations kill the classes outside the rectangle. The matrices of \eqref{cl:rectangle-shift} and \eqref{cl:odd-euler-matrix} are partial permutation matrices in every degree. Their nonzero invariant factors are units, and homogeneous free-module shifts do not change that assertion. \end{proof} For even orthogonal spaces, let \(\dim V=2A\) in \eqref{cl:rank-chain}. We use \(X_j\) only for \(j\geq1\), with the resolutions \(Y_r\) of \eqref{cl:rank-resolution} for \(r\leq R-1\). The resolved normal slices are \(\operatorname{Hom}(\mathbb C^{R-j},\mathcal U_{r-j})\) over \(\operatorname{OG}(r-j,2(A-j))\). Their stable difference \(A-r\) is at least \(1\), so these orthogonal Grassmannians are nonmaximal and connected. \begin{lemma}\label{cl:even-euler} Let \(a>k\geq1\) and \(z=c_k(\mathcal U_k)\). Every graded multiplication map by a positive power of \(z\) on \(H^*(\operatorname{OG}(k,2a);\Lambda)\) has saturated image. \end{lemma} \begin{proof} Put \(c=a-k\). Remove the same homogeneous free factor of rank \(2^{k-1}\) as in the preceding proof. The remaining Borel algebra is \begin{equation}\label{cl:even-borel} C=\frac{\Lambda[e_1,\ldots,e_k,z,w]} {(z^2-e_k,\ w^2-(-1)^ch_c,\ zw,\ h_{c+1},\ldots,h_{c+k-1})}. \end{equation} Here the \(e_i,h_i\) are symmetric functions in squared roots, and \(w\) is the Euler class of the residual \(D_c\) factor. The relations imply \[ e_kh_c=0,\qquad zh_c=0,\qquad we_k=0. \] Modulo the intervening complete symmetric functions, \(e_kh_c=0\) is equivalent, up to sign, to \(h_{c+k}=0\). Let \(B=H^*(\operatorname{Gr}(k,a);\Lambda)\), again with doubled degrees. Reducing words in \(z,w\) in \eqref{cl:even-borel} gives the actual module decomposition \begin{equation}\label{cl:even-module} C=B\oplus zB/(h_c)\oplus wB/(e_k), \end{equation} with \[ B/(h_c)\cong H^*(\operatorname{Gr}(k,a-1);\Lambda), \qquad B/(e_k)\cong H^*(\operatorname{Gr}(k-1,a-1);\Lambda). \] The three displayed annihilator relations are exactly the relations needed to make \eqref{cl:even-module} well-defined. Multiplication by \(z\) sends \(B\) to \(zB/(h_c)\) by the quotient map, sends \(zB/(h_c)\) to \(B\) by multiplication by \(e_k\), and kills the \(w\)-summand. In Schur bases the quotient deletes the partitions touching the removed boundary of the rectangle, while multiplication by \(e_k\) shifts by a full column. Every positive power is therefore a partial permutation matrix. Restoring the homogeneous free factor proves the assertion in all degrees. \end{proof} \begin{proposition}\label{cl:orthogonal-parity} In odd dimension, \(IC(X_0)\) and \(IC(X_1)\) have their ordinary characters as parity objects over every field of odd characteristic. In even dimension the same is true of \(IC(X_j)\) for every \(j\geq1\). Their normalized full-flag pullbacks are actual \(p\)-independent canonical objects. \end{proposition} \begin{proof} All fibers of the smooth projective resolutions above have free integral cohomology in even degrees. Lemma~\ref{cl:resolved-slice} and its even-dimensional version identify all local forms with the Euler maps, whose ranks are unchanged over every odd field by Lemmas~\ref{cl:odd-euler} and \ref{cl:even-euler}. The local intersection-form decomposition criterion \cite[Theorem 3.13]{JMW}, applied successively to the simply connected rank strata, therefore gives the same indecomposable summands and graded characters as in characteristic zero. In odd dimension the first-defect support \(X_1\) occurs already in the resolution of \(X_0\): on its generic normal fiber \(\operatorname{OG}(1,2c+3)\), multiplication by the tautological Euler class has nonzero rank. Alternatively it can be obtained from its own image resolution. In even dimension we use the image resolution of \(X_1\) directly, including when \(A=R\). The smooth full-flag projection has even flag-variety fibers, so normalized pullback preserves the character assertion. \end{proof} \subsection{The pure two-row blocks} We identify the family and a nonzero matrix block of each IC summand. \begin{lemma}\label{cl:odd-stalks} For the odd orthogonal variety \(X_0\), the total unshifted IC stalk dimension at radical defect \(k\) is \[ f_B(k)= \begin{cases}1,&k=0,\\2^{k-1},&k\geq1.\end{cases} \] For the even orthogonal variety \(X_1\), it is \[ f_D(k)= \begin{cases}0,&k=0,\\(k+1)2^{k-2},&k\geq1.\end{cases} \] These formulas are independent of the stable difference. \end{lemma} \begin{proof} We derive the formulas from the smooth image resolutions. Let \(M_B(a,K,r)\) be the total rank of multiplication by \(z^K\) on \(H^*(\operatorname{OG}(r,2a+1))\), with \(M_B(a,K,0)=1\). For \(r\geq1\), the two-summand calculation above and \eqref{cl:rectangle-shift} give \begin{align} M_B(a,2m,r)&=2^r\binom{a-m}{r},\label{cl:odd-ranks}\\ M_B(a,2m+1,r)&= 2^{r-1}\left(\binom{a-m}{r}+\binom{a-m-1}{r}\right). \nonumber \end{align} Here \(\binom nr=0\) unless \(0\leq r\leq n\). Indeed the number of Schur classes surviving a shift by \(m\) full columns is \(\binom{a-m}{r}\). Fix radical dimension \(K\), and let \(F_j(A,K)\) be the stalk dimension at \(Z\) of the rank-\(j\) image \(X_{K-j}\); then \(F_0=1\). Taking the stalk of each smooth image resolution at \(Z\), and using the local-form multiplicities, gives \begin{equation}\label{cl:stalk-recurrence} 2^r\binom Ar =\sum_{j=0}^r M_B(A-j,K-j,r-j)\,F_j(A,K), \qquad 0\leq r\leq K. \end{equation} Consequently \[ (1+2t)^A= \sum_{j=0}^K F_jt^j (1+2t)^{A-\lceil(K+j)/2\rceil} (1+t)^{(K-j)\bmod2}\pmod {t^{K+1}}. \] After division by \((1+2t)^{A-K}\), both sides are polynomials of degree at most \(K\), so this is an exact identity. Set \[ u=\frac{t}{\sqrt{1+2t}},\qquad q=\sqrt{1+u^2}. \] Then \(\sqrt{1+2t}=q+u\), and the identity becomes \begin{equation}\label{cl:stalk-generating} (q+u)^K=\sum_{j=0}^K F_j u^j q^{(K-j)\bmod2}, \qquad q^2=1+u^2. \end{equation} Expanding the left-hand side and reducing powers of \(q\) gives \[ F_j=\sum_{\substack{0\leq h\leq j\\h\equiv j\ (2)}} \binom Kh \binom{\lfloor(K-h)/2\rfloor}{(j-h)/2}. \] For \(j=K>0\) this is the sum of the binomial coefficients of one parity, namely \(2^{K-1}\). The resolved-slice calculation applies the same formula at every defect, proving \(f_B\). For even orthogonal Grassmannians and positive \(K\), the total rank is \begin{align} M_D(a,2m,r)&= 2^{r-1}\left(\binom{a-m}{r}+\binom{a-m-1}{r}\right), \label{cl:even-ranks}\\ M_D(a,2m+1,r)&=2^r\binom{a-m-1}{r}.\nonumber \end{align} This is \(M_B(a,K+1,r)\). The recurrence for the even rank images through \(r\leq K-1\) is therefore the odd recurrence with domain size \(K+1\), through the same ranks. The preceding coefficient formula gives \[ F_{K-1}(A,K+1) =\frac12\sum_{\substack{0\leq h\leq K+1\\h\equiv K+1\ (2)}} (K+1-h)\binom{K+1}{h} =(K+1)2^{K-2}. \] For \(K=1\) the same formula is \(F_0=1\). This proves the asserted \(X_1\) stalk formula. Only nonmaximal orthogonal Grassmannians occur. The recurrence is used only through rank \(K-1\), and its displayed coefficient is independent of \(A\); the same formula therefore applies when \(A=R\). \end{proof} \begin{proposition}\label{cl:pure-two-row} There are actual sources at every odd coefficient characteristic with the following standard blocks and exact special families: \[ \begin{array}{c|c|c|c} K&\text{range}&\text{standard block}&\text{special family member}\\ \hline B_{A+R}&A\geq R\geq1& (\varnothing;\,1^A+1^R)&(1^R;\,1^A)\\ D_{A+R}&A\geq R\geq2& \{1^A+1^R;\,\varnothing\}& \{(2,1^{A-1});\,1^{R-1}\}. \end{array} \] Here sums of partitions are rowwise. The corresponding positive blocks are \(E_{((A,R),\varnothing)}\) and \(E_{\{(A,R),\varnothing\}}\), respectively. \end{proposition} \begin{proof} In odd dimension use the full-flag pullback of \(IC(X_0)\). With \(c=A-R\), and with the convention that \(s_0\) changes the first coordinate sign, its maximal full-flag label is \[ w=w_0^{B_{A+R}}u,\qquad u=(1,\ldots,c,A+1,\ldots,A+R,c+1,\ldots,A). \] Set \(J=B_A\times\mathfrak S_R\), and use the isotropic coordinate flag for which \(Z=\langle e_{A+1},\ldots,e_{A+R}\rangle\). The corresponding \(J\)-minimal representative is \[ w^J=(1,\ldots,c,A+1,\ldots,A+R; -A,\ldots,-(c+1)), \qquad w=w^Jw_J. \] Its last \(R\) coordinates select an isotropic plane in the nondegenerate summand of \(Z^\perp\), so its partial-flag Borel orbit lies in \(S_0\). The displayed representative is \(J\)-minimal: its first \(A\) entries are positive and increasing, and its last \(R\) entries are increasing in signed order. The only inversions in \(u\) are between its two blocks of size \(R\), so \(\ell(u)=R^2\). Hence \[ \begin{split} \ell(w^J) &= (A+R)^2-R^2 -\left(A^2+\frac{R(R-1)}2\right)\\ &=2AR-\frac{R(R-1)}2 =\dim X_0. \end{split} \] The irreducibility of \(X_0\) follows also from its birational image resolution. Thus the indicated Borel orbit is dense in \(X_0\), proving that \(w^J\), and therefore \(w\) after full-flag pullback, is its actual Schubert label. Row insertion of \(u\) gives the two-row shape \((A,R)\): the initial small block and the large block form the first row, and the last middle block bumps exactly \(R\) large entries. The special type-\(A\) representation is \(S^{(A,R)}\), of minimum degree \(R\). Its truncated induction to \(B_{A+R}\) is the special bipartition \(((A),(R))\): the horizontal Pieri rule exhibits this constituent and it is the unique one of that degree. Ordinary parabolic cell induction, followed by multiplication by \(w_0\), therefore gives the special family member \((1^R;1^A)\). In particular its \(a\)-value is \(A^2+R(R-1)\). At \(v=1\), map the signed-subset spherical module to the unsigned \(R\)-subset module by killing the normal sign subgroup. For an unsigned subset with \(k\) radical positions, the condition \(U\subset Z^\perp\) forces the signs of those positions; the remaining \(R-k\) signs are free. Lemma~\ref{cl:odd-stalks} gives coefficient \[ 2^{R-k}f_B(k)=2^{R-1}(1+\mathbf1_{\{k=0\}}). \] A harmonic polytabloid for \(S^{(A,R)}\) is the alternating sum over the \(2^R\) choices of one element from each of \(R\) disjoint top-bottom pairs. Choose each top element outside the radical block and each bottom element inside; \(A\geq R\) ensures that this is possible. Give a choice with \(k\) bottom elements the sign \((-1)^k\). Its pairing with the constant function is \((1-1)^R=0\), and its pairing with \(\mathbf1_{\{k=0\}}\) is \(1\). The block action therefore has nonzero pairing \(2^{R-1}\). In even dimension use \(IC(X_1)\), with parabolic \(J=D_A\times\mathfrak S_R\). Put \(n=A+R\) and \(\delta=(-1)^n\). The full-flag signed permutation is \begin{equation}\label{cl:even-word} w=(\delta,-2,\ldots,-(c+1),-(A+2),\ldots,-n, A+1,-(c+2),\ldots,-A). \end{equation} It is even-signed. To see that it is the required geometric label, the partial-flag Schubert label is \[ ((-1)^{R+1},2,\ldots,c+1,A+2,\ldots,n;\ -A,\ldots,-(c+2),A+1). \] Its final \(R\) entries select the \(R-1\) largest negative nondegenerate coordinates and the first radical coordinate, which are exactly the incidence conditions defining \(X_1\). Multiplication by \(w_J\) gives \eqref{cl:even-word}. For all \(A\geq R\geq2\), denote the displayed partial-flag representative by \(w^J\), so that \(w=w^Jw_J\). Its first \(A\) entries have no \(D_A\)-descent, and its last \(R\) entries are increasing, so it is \(J\)-minimal. Its selected isotropic plane lies in \(Z^\perp\) and has exactly one radical coordinate, hence its Borel orbit lies in \(S_1\). In our signed-permutation convention the longest element of \(D_n\) is \(((-1)^{n-1},-2,\ldots,-n)\). Consequently, for either parity of \(n\), \[ v=w_0^{D_n}w =(-1,2,\ldots,c+1,A+2,\ldots,n, -(A+1),c+2,\ldots,A). \] For an even signed permutation, the length is the number of inversions in signed order plus the number of pairs of entries with negative sum. In this word those two numbers are respectively \[ (R-1)^2+(R-1)+c+1, \qquad 1+c+(R-1)=A. \] Thus \(\ell(v)=R^2+2c+1\), and \[ \begin{split} \ell(w^J) &=n(n-1)-(R^2+2c+1) -\left(A(A-1)+\frac{R(R-1)}2\right)\\ &=2A(R-1)-\frac{(R-1)(R-2)}2. \end{split} \] The right-hand side is \(\dim Y_{R-1}=\dim X_1\): the base \(\operatorname{OG}(R-1,2A)\) has dimension \(2A(R-1)-\bigl(3(R-1)^2+(R-1)\bigr)/2\), and the ordinary Grassmannian fiber has dimension \(R(R-1)\). Since \(X_1\) is irreducible, the indicated Borel orbit is dense in it. This proves the actual Schubert-label assertion, including the boundary case \(A=R\). The doubled word \((-w(n),\ldots,-w(1),w(1),\ldots,w(n))\) has RSK shape \begin{equation}\label{cl:even-rsk} \begin{cases} (4,2^{\,2R-2},1^{\,2c}),&n\text{ even},\\ (3,2^{\,2R-2},1^{\,2c+1}),&n\text{ odd}. \end{cases} \end{equation} To compute its Greene invariants, write its successive blocks as \[ \begin{split} &(A,\ldots,c+2),\ -(A+1),\ (n,\ldots,A+2),\ (c+1,\ldots,2),\\ &-\delta,\delta,\ (-2,\ldots,-(c+1)),\ (-(A+2),\ldots,-n),\ A+1,\ (-(c+2),\ldots,-A). \end{split} \] A decreasing chain avoiding \(\pm(A+1)\) uses at most one of the two positive blocks of length \(R-1\), at most \(2c\) letters from the head blocks, and at most one of the two negative blocks of length \(R-1\). It can use at most one of \(-1,1\) when \(n\) is even, and both when \(n\) is odd. Its length is therefore at most \(2A-1\) or \(2A\), respectively. A chain containing \(-(A+1)\) can use only the first positive block before it and the block \(-(A+2),\ldots,-n\) after it. A chain containing \(A+1\) has the symmetric bound. Such a chain has length at most \(2R-1\). When \(n\) is even, the two decreasing chains \[ \begin{gathered} (A,\ldots,2,-1,-2,\ldots,-A),\\ (n,\ldots,A+2,1,-(A+2),\ldots,-n) \end{gathered} \] cover all letters except \(\pm(A+1)\), and the first has length \(2A-1\). The increasing chain \((-(A+1),-1,1,A+1)\) forces two decreasing chains to omit at least two letters, and three to omit at least one. Adding either special letter as a singleton gives a three-chain cover of \(2n-1\) letters; adding both gives a four-chain cover of the word. The maximal sizes of unions of one, two, three and four decreasing chains are therefore \(2A-1,2n-2,2n-1,2n\). When \(n\) is odd, the decreasing chains \[ \begin{gathered} (n,\ldots,A+2,c+1,\ldots,1,-1,\ldots,-A),\\ (A,\ldots,c+2,-(A+1),-(A+2),\ldots,-n) \end{gathered} \] have lengths \(2A\) and \(2R-1\), and cover all letters except \(A+1\). The increasing chain \((-(A+1),1,A+1)\) prevents a two-chain cover of the entire word. Adding \(A+1\) as a singleton gives a three-chain cover. The maximal union sizes are \(2A,2n-1,2n\). Greene's theorem \cite[Theorem 2.14]{BXX} gives conjugate RSK partitions \((2A-1,2R-1,1,1)\) and \((2A,2R-1,1)\), respectively, which are equivalent to \eqref{cl:even-rsk}. Apply the type-\(D\) symbol construction to this doubled word \cite[\S3.3, Propositions 3.5 and 3.6]{BXX}. Pad its RSK partition \(\nu\) with zeros to length \(2A\). In the defect-zero normalization, the union of the two symbol rows is \[ \left\{\left\lfloor\frac{\nu_i+2A-i}{2}\right\rfloor: 1\leq i\leq2A\right\}. \] For either parity this multiset is the union of \[ (1,2,\ldots,A-1,A+1),\qquad (0,1,\ldots,c,c+2,\ldots,A). \] Indeed the first row of \(\nu\) contributes \(A+1\). Its \(2R-2\) rows of length two contribute \(A\) and \(c+1\) once, and each intervening integer twice. The remaining rows contribute \(0\) once and \(1,\ldots,c\) twice. Sorting the symbol gives the special label \(\{(2,1^{A-1});1^{R-1}\}\), whose \(a\)-value is \(A(A-1)+(R-1)^2\). The same sign count gives the unsigned-subset coefficient of the even IC source: \[ 2^{R-k}f_D(k)=2^{R-2}\bigl(k+1-\mathbf1_{\{k=0\}}\bigr). \] For \(R\geq2\), the same harmonic polytabloid kills the linear function \(k+1\), since \[ \sum_{k=0}^R(-1)^k\binom Rk(k+1) =(1-1)^R-R(1-1)^{R-1}=0. \] Its pairing with the indicator is \(1\), so the block action is \(-2^{R-2}\neq0\). Proposition~\ref{cl:orthogonal-parity} and Lemma~\ref{cl:matrix-block} now prove both assertions. \end{proof} \subsection{Column formulas and the common discriminant} The following elementary column function will organize the construction: \begin{equation}\label{cl:pair-function} c(a,b)= \begin{cases} (2a-1,2b+1),&a>b,\\ (2b,2a),&a\leq b. \end{cases} \end{equation} Zero columns are omitted. We call the two cases respectively the \(D\)-orientation and the \(B\)-orientation of a pair; these names describe the formula, not the type of the ambient local factor. \begin{lemma}\label{cl:distinguished-columns} The trivial-local-system Springer representations and their orbit columns have the following descriptions. \begin{enumerate} \item In type \(C\), a distinguished bipartition satisfies \[ \alpha_i\geq\beta_i-1,\qquad \beta_i\geq\alpha_{i+1}-1. \] If \(a_j=\alpha'_j\) and \(b_j=\beta'_j\), the conjugate Springer partition is the decreasing concatenation \[ c(a_1,b_1),c(a_2,b_2),\ldots . \] \item In type \(B\), the distinguished inequalities are \(\alpha_i\geq\beta_i-2\) and \(\beta_i\geq\alpha_{i+1}\). The orbit columns are \[ (2b_1+1,\ c(a_1,b_2),c(a_2,b_3),\ldots). \] The minimum polynomial has the initial factor \(B_{b_1}\). \item In type \(D\), orient the unordered distinguished label so that \(\alpha_i\geq\beta_i\) and \(\beta_i\geq\alpha_{i+1}-2\). The orbit columns are \[ (2a_1,\ c(a_2,b_1),c(a_3,b_2),\ldots). \] The minimum polynomial has initial factor \(D_{a_1}\). \end{enumerate} \end{lemma} \begin{proof} We spell out the column count in the Springer partition formulas \cite[\S2]{AHS}, \cite[\S\S5.2, 7.2--7.3]{JLS}. In type \(C\), interleave the even rows \(2\alpha_1,2\beta_1,2\alpha_2,2\beta_2,\ldots\). Distinguishedness says that an ascending adjacent pair has the form \((2j-2,2j)\); replace it by \((2j-1,2j-1)\). Before replacement the two columns \(2j-1,2j\) both have height \(a_j+b_j\). The indicator strings for \(\alpha_i\geq j\) and \(\beta_i\geq j\) contain \(a_j\) and \(b_j\) initial ones. Their interleaving has \(b_j-a_j\) upward transitions if \(b_j\geq a_j\), and \(a_j-b_j-1\) otherwise. Each replacement raises the odd column by one and lowers the even column by one. This gives exactly \eqref{cl:pair-function}. The distinguished inequalities, equivalently \(a_j\geq b_{j+1}\) and \(b_j\geq a_{j+1}-1\), show that the concatenation is decreasing. For type \(B\), remove the first column of \(\beta\). The pair \((\alpha,(\beta_i-1)_+)\) satisfies the type-\(C\) inequalities. Counting columns in the interleaved rows \((2\alpha_i+1,2\beta_i-1)\), with ascending pairs averaged, adds the initial column \(2b_1+1\) to the type-\(C\) tail. For type \(D\), remove the first column of \(\alpha\) instead; the same count in \((2\alpha_i-1,2\beta_i+1)\) adds \(2a_1\). The factorizations of \eqref{cl:signed-specht} give the claimed head factors. \end{proof} The tensor calculation will always use the same elementary observation. \begin{lemma}\label{cl:common-discriminant} Let \(L\) be a parabolic subgroup of \(M=W_y\), occurring among the \(p\)-th-root stabilizers in Proposition~\ref{op:tensor}. Suppose that the completed ideals of a finite collection of actual source families contain \(\Delta_Lh_1,\ldots,\Delta_Lh_m\), respectively, with every \(h_i\in S_y^L\). The completed coefficient ideal of their tensor product and block closure contains \[ \Delta_L\prod_{i=1}^m h_i . \] Its support bound is the intersection of the bounds of the source families. \end{lemma} \begin{proof} Use the term with \(L_t=L\) in formula~\eqref{op:tensor-formula}. It sends \(\Delta_Lh_i\) and \(\Delta_Lh_{i+1}\) to \(\Delta_Lh_ih_{i+1}\). Iteration proves the first assertion. The corresponding operations are tensor products and block projections of actual objects, so the support assertion follows from the support interface and Theorem~\ref{lift:whole}. \end{proof} \begin{lemma}\label{cl:root-sheets} Let \(y\) be a full-rank torus point in type \(B_d,C_d\) or \(D_d\), and let \(p>h\). Each common parabolic \(L\) displayed in Propositions~\ref{cl:c-generation}, \ref{cl:b-generation} and \ref{cl:d-generation-nonvery} is \(W_t\) for some \(t\) with \(t^p=y\). \end{lemma} \begin{proof} The vector coordinates of \(y\) are \(\epsilon_i\in\{1,-1\}\). The displayed groups consist of symmetric coordinate blocks and, in orthogonal type, retained plus and minus heads of sizes \(r,s\). Assign to each symmetric block a common value \(z\) satisfying \(z^p=\epsilon_i\). For blocks of the same sign, choose values from distinct pairs \(\{z,z^{-1}\}\), excluding \(z=\epsilon_i\). There are \((p-1)/2\) such pairs for each sign. Values belonging to different signs are neither equal nor inverse, since their \(p\)-th powers differ. On each retained head put \(z=\epsilon_i\). Thus permutations preserving the chosen values give exactly the prescribed symmetric factors; coordinate flips occur only in the heads. For types \(B_d,C_d\), the inequality \(p>2d\) provides at least \(d\) pairs, enough for all symmetric blocks. For type \(D_d\), the inequality \(p>2d-2\) provides at least \(d-1\) pairs. By Lemma~\ref{cl:distinguished-columns}, every nonempty distinguished local \(D\)-label has a nonempty head. The tail therefore contains at most \(d-1\) symmetric blocks. For \(\operatorname{Sp}_{2d}\), these vector coordinates already define \(t\), with the required stabilizer. For a Spin torus they have two lifts. Multiplication by the nontrivial element in the kernel of \(\operatorname{Spin}\to\operatorname{SO}\) interchanges their \(p\)-th powers because \(p\) is odd. Exactly one lift therefore has \(p\)-th power \(y\). To compute its stabilizer, let \(e_i\) be the \(i\)-th vector coordinate character and use the spin weight \(\omega=(e_1+\cdots+e_d)/2\). A coordinate permutation fixes \(\omega\). Flipping coordinates in a set \(I\) sends it to \(\omega-\sum_{i\in I}e_i\), changing its value by \(\prod_{i\in I}z_i^{-1}\). Generic tail values exclude coordinate flips. In the heads this product is \((-1)^{|I_-|}\), where \(I_-\) is the set of flipped minus-head coordinates. In odd orthogonal type the lift stabilizer is consequently \(B_r\times D_s\) on the heads. In even orthogonal type the global even-sign condition also forces an even number of plus-head flips, giving \(D_r\times D_s\). Together with the symmetric factors these are precisely the stated groups \(L\). \end{proof} If \(L_j=\mathfrak S_{a+b}\) permutes the variables in one column pair, the alternation of \(D_a(I)B_b(J)\) is nonzero. Indeed its alternant exponents are \[ \{0,2,\ldots,2a-2\}\ \cup\ \{1,3,\ldots,2b-1\}, \] which are all distinct. The alternant is divisible by the ordinary Vandermonde \(\Delta_{L_j}\), with invariant quotient. \subsection{Symplectic sources at the full-rank points} For \(G=\operatorname{Sp}_{2d}\), a full-rank torus point has stabilizer \(M=B_u\times B_v\), on its \(+1\) and \(-1\) coordinate blocks. Choose distinguished local types \((\alpha_+,\beta_+)\) and \((\alpha_-,\beta_-)\). For one column position, abbreviate the heights to \((a_+,b_+)\) and \((a_-,b_-)\). Write \[ (X,Y)=c(a_+,b_+)+c(a_-,b_-),\qquad m=(X+Y)/2, \] where pairs are added componentwise. \begin{lemma}\label{cl:c-pair-source} There is an actual minuscule finite affine-parabolic source on these \(m\) coordinates whose exact symplectic orbit has columns \((X,Y)\), and whose local coefficient ideal contains the minimum copy of \[ ((1^{a_+}),(1^{b_+})) \boxtimes ((1^{a_-}),(1^{b_-})). \] \end{lemma} \begin{proof} For a standard two-column \(B\)-block \((1^a;1^b)\), the exact family orbit in symplectic type has columns \begin{equation}\label{cl:b-family-columns} \begin{cases} (2b,2a),&a\leq b,\\ (2a-1,2a-1),&a=b+1,\\ (2a-2,2b+2),&a\geq b+2. \end{cases} \end{equation} In the first two cases the finite label is special. In the last case its sorted defect-one symbol has special representative \((1^{b+1};1^{a-1})\); Lemma~\ref{cl:distinguished-columns} then gives the third line. A special two-column \(D\)-block \(\{1^a;1^b\}\) gives \((2a-1,2b+1)\) when \(a>b\), and \((2a,2a)\) when \(a=b\), by \eqref{cl:finite-label-rule}. The required choices are \[ \begin{array}{c|c} \text{local orientations}&\text{standard finite source block}\\ \hline B,B&B_m:\ (1^{a_++a_-};\,1^{b_++b_-})\\ D,D&B_m:\ (1^{a_++a_-};\,1^{b_++b_-})\\ D,B\text{ or }B,D& D_m:\ \{1^{a_++b_-};\,1^{b_++a_-}\}. \end{array} \] For the first line, the first case of \eqref{cl:b-family-columns} applies. For the second line the two strict inequalities give \(a_++a_-\geq b_++b_-+2\), so the third case applies. For the mixed line the two heights are unequal; the larger height is the one associated with the \(D\)-oriented local pair. Substitution gives \((X,Y)\) in all cases. In the \(B\)-sources, splitting a vertical partition between the two local coordinate blocks has Littlewood--Richardson coefficient one in each color. In the \(D\)-source use the affine subgroup fixing \(\xi=(1/2,\ldots,1/2)\) on its coordinates. At a plus coordinate its finite character is trivial; on the minus coordinates it is the sign-subgroup character, which interchanges the two colors. Thus the global ordered extension with heights \((a_++b_-,b_++a_-)\), restricted to the whole intersection with \(M\) and twisted by that character, contains the required local type. Both ordered extensions restrict to the same nondegenerate \(D_m\)-representation, so there is no choice of a degenerate half here. The exact finite Mackey formula and Lemma~\ref{cl:matrix-block} give the whole local isotypic ideal, and hence its minimum copy. If \(m=1\), the mixed pair is \((1,0)\) and its minimum polynomial is \(1\); use the identity source. \end{proof} \begin{proposition}\label{cl:c-generation} At every full-rank symplectic torus point, each local trivial-local-system minimum Springer copy \(U_{\mathcal O}\) is contained in the completed ideal of a finite actual source family with support in the ambient orbit closure of \(\mathcal O\). \end{proposition} \begin{proof} Let \(\lambda\) be the row union of the two local symplectic partitions. Its columns are the decreasing list \[ (X_1,Y_1,X_2,Y_2,\ldots),\qquad (X_j,Y_j)=c(a_j^+,b_j^+)+c(a_j^-,b_j^-). \] Put \(m_j=(X_j+Y_j)/2\). For each \(j\), use Lemma~\ref{cl:c-pair-source} on group \(j\), and a symmetric-group longest source on every other group. By the finite label rule its bound \(\sigma_j\) is the symplectic collapse of the partition with columns \begin{equation}\label{cl:c-cut} (m_1,m_1,\ldots,m_{j-1},m_{j-1},X_j,Y_j, m_{j+1},m_{j+1},\ldots). \end{equation} This list is decreasing. Pair averaging decreases column dominance, so its uncollapsed row partition lies above \(\lambda\). Since \(\lambda\) is symplectic, it remains below the collapse. For a partition \(\nu\), the rank of the \(k\)-th power of a nilpotent with partition \(\nu\) is \[ r_k(\nu)=|\nu|-\sum_{i=1}^k\nu'_i. \] Every even column prefix in \eqref{cl:c-cut} equals the corresponding prefix of \(\lambda\), and the odd prefix ending at \(X_j\) equals it for cut \(j\). Collapse can only increase the column prefixes, and cannot cross \(\lambda\). Therefore \begin{equation}\label{cl:c-orbit-meet} \min_j r_k(\sigma_j)=r_k(\lambda)\quad\text{for every }k, \qquad \bigcap_j\overline{\mathcal O_{\sigma_j}} =\overline{\mathcal O_\lambda}. \end{equation} Choose disjoint column coordinate groups and put \[ P_j=D_{a_j^+}B_{b_j^+}D_{a_j^-}B_{b_j^-},\quad L_j=\mathfrak S_{a_j^++b_j^+} \times\mathfrak S_{a_j^-+b_j^-},\quad L=\prod_jL_j . \] Lemma~\ref{cl:root-sheets} realizes \(L\) as a \(p\)-th-root stabilizer. The exact local block inclusion puts \(P_j\) in the active ideal; each passive longest source supplies \(\Delta_{L_i}\). Thus the \(j\)-th ideal contains \[ g_j=\operatorname{Alt}_{L_j}(P_j) \prod_{i\neq j}\Delta_{L_i} =\Delta_L h_j,\qquad h_j\in S_y^L . \] The distinct even and odd exponent sets prove every alternant nonzero. Lemma~\ref{cl:common-discriminant} produces \[ \Delta_L\prod_jh_j =\prod_j\operatorname{Alt}_{L_j}(P_j) =\operatorname{Alt}_L\left(\prod_jP_j\right)\neq0 . \] The product before alternation is the signed column Specht polynomial in the unique minimum copy of the desired local type. Its nonzero group-algebra projection remains in that copy, and \(M\)-saturation gives all of it. The support bound is \eqref{cl:c-orbit-meet}. The construction has one cut for each nonempty pair, and is finite in every rank. In type \(B_u\times B_v\) the corresponding minimum Specht ideal is also the full isotypic ideal \cite[Proposition 7.6]{DMRV}; only the minimum-copy assertion is needed for the general downset argument. \end{proof} \subsection{Orthogonal mixed points and complete intersection groups} We next treat the simply connected orthogonal groups. Their full-rank centralizers have Weyl groups \[ B_u\times D_v\quad\text{for }\operatorname{Spin}_{2d+1}, \qquad D_u\times D_v\quad\text{for }\operatorname{Spin}_{2d}. \] Their vector torus coordinates are \(+1\) and \(-1\). All lifts to the Spin torus will be retained. The head-and-pair formulas in Lemma~\ref{cl:distinguished-columns} reduce the construction to local heads and one pair at a time. We make explicit a point about the finite intersections. An affine parabolic at the half-shift end fixes a rational vector \(\xi\), and its constant character on an element \(h\) fixing the torus point is \(y^{\xi-h\xi}\). When coordinates are split into two global ends and two local signs, the intersection need not be a product of four even-sign groups. It contains coupled odd sign changes between quadrants. We shall compute restrictions in the product of the four hyperoctahedral overgroups, twist there by the sign character in the half-shift minus quadrant, and only then restrict to the whole intersection. This retains the value on every coupled element. \begin{lemma}\label{cl:orthogonal-retention} For the active blocks specified below, vertical-column Littlewood--Richardson restriction, with the indicated half-shift character, puts the desired local head-and-pair minimum polynomial in the source ideal. Passive symmetric-group longest factors supply the ordinary column discriminants with trivial cocycle. For a balanced local type-\(D\) label, the initial assertion is occurrence of at least one nonzero minimum-half component; the individual halves are treated in Lemma~\ref{cl:half-switch}. \end{lemma} \begin{proof} The exact finite-block coefficient formula takes the restriction of the global block to the full intersection, twists by \(y^{\xi-h\xi}\), and induces to the local stabilizer. A common constituent of those restrictions therefore supplies the entire corresponding local isotypic ideal, by the matrix-block calculation. On the four hyperoctahedral overgroups, the character is the sign of coordinate flips in the quadrant which lies both at the half-shift end and in the minus block. Tensoring by this character exchanges its two partition colors. The vertical Littlewood--Richardson constituents listed below thus give the desired restrictions before passing to the full intersection, so they also give them after restriction. In particular their character on a coupled pair of odd flips is retained. Each passive symmetric block has constant \(\xi\), hence \(\xi-h\xi=0\), and its determinant restricts to the ordinary determinant of its intersection. Its coefficient image therefore contains the corresponding discriminant. Adjoining the remaining passive column pairs preserves the required full local constituent: in each color the rowwise sum of its vertical column partitions is the desired partition, and the corresponding Cartan Littlewood--Richardson coefficient is one. On an unsigned pair group the determinant occurs in the restriction of the required two-column type, again with coefficient one. Thus ordinary support containment applies to the complete target representation selected by the finite Mackey formula. \end{proof} \subsection{Odd orthogonal head-and-pair sources} Write a plus-side distinguished head and pair as \((r;a,b)\), and a minus-side distinguished head and pair as \((s;c,d)\). Thus \[ F_+=(1^a;\,1^r+1^b),\qquad F_-=\{1^s+1^c;\,1^d\}, \qquad r\geq a-1,\ r\geq b,\ s\geq c,d . \] The local row union has columns \begin{equation}\label{cl:odd-head-pair} H=2(r+s)+1,\qquad (X,Y)=c(a,b)+c(c,d). \end{equation} The affine Coxeter diagram has two finite \(B\)-ends. For \(P\in\{r,b\}\), \(Q\in\{s,c\}\), use the standard blocks \begin{equation}\label{cl:odd-pair-blocks} E_1=(1^{a+Q};\,1^{P+d}),\qquad E_2=(\varnothing;\,1^{r+b-P+s+c-Q}) \end{equation} on the untwisted and half-shift ends respectively. The first is minuscule and the second is longest. The special representative of \(E_1\) is \[ (1^u;1^v),\qquad u=\min(a+Q,P+d+1),\quad v=\max(P+d,a+Q-1). \] Writing \(z=r+b-P+s+c-Q\), truncated induction gives \((1^u;1^v+1^z)\). \begin{lemma}\label{cl:odd-active} For every choice of local orientations there is an actual active source whose bound contains the target in \eqref{cl:odd-head-pair} and whose two largest column prefix is \(H+X\). \end{lemma} \begin{proof} For orientations \(B,B\), choose \(P=b,Q=c\) in \eqref{cl:odd-pair-blocks}. Then \(z=r+s\), \(a+c\leq b+d\), and \(r+s\geq b+d\). The induced label has columns \((H,2(b+d),2(a+c))=(H,X,Y)\). For orientations \(D,D\), make the same choice. Now \(a+c\geq b+d+2\), so \(u=b+d+1\), \(v=a+c-1\), and \(z=r+s\geq v\). The columns are \((H,2(a+c)-2,2(b+d)+2)=(H,X,Y)\). For orientations \(D,B\), take \(P=r,Q=s\). Here \(z=b+c\) and both \(a+s\) and \(r+d+1\) are at least \(b+c+1\). Thus \(u\geq z+1\) and \(v\geq u-1\). Its final column is \(2z+1=Y\), so its first two columns sum to \(H+X\). The first column need not itself be \(H\), which will be supplied by a separate head cut. The remaining orientation \(B,D\) uses Proposition~\ref{cl:pure-two-row}. Put \[ P_0=r+s,\qquad R_0=b+c, \] and use \begin{equation}\label{cl:odd-pure-active} E_1=(1^a;1^d),\qquad E_2=(\varnothing;\,1^{P_0}+1^{R_0}) \end{equation} on the two ends. The inequalities imply \(P_0\geq R_0\). If \(u=\min(a,d+1)\) and \(v=\max(d,a-1)\), their special representatives truncate-induce to \[ (1^{R_0}+1^u;\,1^{P_0}+1^v). \] Since \(P_0\geq R_0>u,v\) and \(v\geq u-1\), this is distinguished. Its columns are \[ (2P_0+1,\ 2R_0-1,\ 2v+1,\ 2u-1,\ 1), \] with the last two entries omitted when \(u=0\). The first two entries are \(H,X\), and the remainder has total \(Y=2(a+d)+1\). We verify the local occurrence, which also supplies the lower orbit inequality. In \eqref{cl:odd-pair-blocks}, split the plus beta columns into \(P,r+b-P\), and the minus alpha columns into \(Q,s+c-Q\). The remaining plus alpha and minus beta columns lie in the first end. Vertical restriction has coefficient one, and the half-shift minus twist turns the second minus beta column into the required alpha column. For \eqref{cl:odd-pure-active}, the required coefficient is the transpose-Cartan coefficient \[ c_{\,1^r+1^b,\ 1^s+1^c}^{\,1^{r+s}+1^{b+c}}=1 . \] The half-shift twist again changes the minus color, and \(E_1\) supplies its remaining two columns. Lemma~\ref{cl:orthogonal-retention} gives the full intersection occurrence in all four cases. The constant matrix-block argument in the proof of Proposition~\ref{ord:containment}, applied to these selected local blocks, forces the target orbit to lie in each source bound. \end{proof} \begin{proposition}\label{cl:b-generation} At every full-rank torus point for \(\operatorname{Spin}_{2d+1}\), every local trivial-local-system minimum Springer copy has a finite actual source family with the exact ambient orbit bound. \end{proposition} \begin{proof} Write \(F_+,F_-\) for the full local Springer labels. Use Lemma~\ref{cl:odd-active} for each tail-pair position \(j\), and symmetric-group longest factors on all other positions. Add a head source: the ordinary \(B_{r+s}\) longest object on the combined heads, with the same passive symmetric factors. The common local parabolic is \[ L=B_r\times D_s\times \prod_j\bigl( \mathfrak S_{a_j+b_j}\times \mathfrak S_{c_j+d_j}\bigr). \] This is a \(p\)-th-root stabilizer by Lemma~\ref{cl:root-sheets}. The head source is ordinary, with fixed point \(\xi=0\), so its character is untwisted and its local image contains \(\Delta_L\). If an active minus-side label is balanced, apply the odd-orthogonal half switch described after Lemma~\ref{cl:half-switch} to that source ideal first. Choose the flipped coordinate in its head, so all passive coordinates stay fixed. Central stability gives both minimum halves at the same bound, and hence the unprojected active minimum polynomial. The \(j\)-th active source contains \(\Delta_Lh_j\), where \(h_j\) is obtained by alternating \(D_{a_j}B_{b_j}\) and \(D_{c_j}B_{d_j}\) within their pair groups, and dividing by those two Vandermondes. The alternants are nonzero by their distinct even and odd exponent sets. Lemma~\ref{cl:common-discriminant}, including the head factor \(1\), gives \[ \Delta_L\prod_jh_j =\operatorname{Alt}_L(P_{F_+}P_{F_-})\neq0 . \] This is a nonzero vector in the target minimum copy, and \(M\)-saturation gives the copy. Let the target columns be \((H,X_1,Y_1,X_2,Y_2,\ldots)\). The head source gives every odd prefix: its head and its preceding averaged pairs have the same sum as the target prefix. For the even prefix ending at \(X_j\), use the \(j\)-th active source; its first two active columns sum to \(H+X_j\), and the passive preceding pairs retain their totals. These selected columns are a subset of the appropriate size in the raw source partition, so the largest columns have at least that sum. Orthogonal collapse can only increase it. Conversely the full-intersection occurrence of the target in every source, including the passive factors, and Proposition~\ref{ord:containment}, give the opposite inequality. Thus at each prefix one source attains the target exactly. The rank functions consequently have pointwise minimum equal to that of the target, and the intersection of the source closures is its orbit closure. If a local \(D\)-type is degenerate, an odd coordinate flip in that local block is a global \(B\)-Weyl element. It fixes the vector torus values and may change the Spin lift by the central element. The central stability proved in Lemma~\ref{cl:central-stability} returns to the same Spin point and exchanges the two halves. Their ambient odd orthogonal orbit is the same. Thus both required halves are supplied at the same bound. \end{proof} \subsection{Even orthogonal head-and-pair sources} At a full-rank mixed point of \(\operatorname{Spin}_{2d}\), write \[ F_+=\{1^r+1^a;\,1^b\},\qquad F_-=\{1^s+1^c;\,1^d\},\qquad r\geq a,b,\quad s\geq c,d,\quad P=r+s . \] The target head is \(H=2P\), and its tail pair is \((X,Y)=c(a,b)+c(c,d)\). Use the two \(D\)-ends of the affine diagram. A two-column block means a minuscule source from Proposition~\ref{cl:minuscule}; when it is balanced retain the two actual fork sources at this stage. \begin{lemma}\label{cl:even-active} The following standard active blocks supply a nonzero target local minimum and a source bound whose first two column prefix is \(H+X\): \[ \begin{array}{c|c|c} \text{orientations}&\text{untwisted end}&\text{half-shift end}\\ \hline B,B&\{1^P;1^{b+d}\}&\{1^a;1^c\}\\ D,B&\{1^P;1^b\}&\{1^{a+d};1^c\}\\ B,D&\{1^P;1^d\}&\{1^a;1^{b+c}\}\\ D,D&\{1^P+1^{a+c};\varnothing\}&\{1^d;1^b\}. \end{array} \] For a balanced local type this initially supplies at least one nonzero half. The individual required halves are obtained in Lemma~\ref{cl:half-switch} and Proposition~\ref{cl:very-even}. \end{lemma} \begin{proof} Only the last row uses a new source. There \(R=a+c\geq2\) and \(P\geq R\), so Proposition~\ref{cl:pure-two-row} supplies the first block. Its special representative is \(\{(2,1^{P-1});1^{R-1}\}\). For the intersection calculation, split the coordinates into quadrants \(11,12,21,22\), with the first index specifying the global end and the second the local sign. In the product of their hyperoctahedral overgroups use the following ordered bipartitions: \[ \begin{array}{c|c|c|c|c} &11&12&21&22\\ \hline B,B&(1^r;1^b)&(1^s;1^d)&(1^a;\varnothing)& (\varnothing;1^c)\\ D,B&(1^r;1^b)&(1^s;\varnothing)&(1^a;\varnothing)& (1^d;1^c)\\ B,D&(1^r;\varnothing)&(1^s;1^d)&(1^a;1^b)& (\varnothing;1^c)\\ D,D&(1^r+1^a;\varnothing)&(1^s+1^c;\varnothing)& (\varnothing;1^b)&(1^d;\varnothing). \end{array} \] Vertical-column restriction gives the two global-end representations in each row. In the last row the nontrivial coefficient is \[ c_{\,1^r+1^a,\ 1^s+1^c}^{\,1^{r+s}+1^{a+c}}=1 . \] Exchange the colors in quadrant \(22\), as required by the half-shift character. The local columns then assemble to \(F_+\) and \(F_-\). Restricting these common overgroup constituents to the whole actual intersection proves the desired occurrence, including the coupled odd flips. When a local unordered type is balanced, take a nonzero projection onto one of its two halves; occurrence of the other half is not inferred merely by projection. We compute the source bounds from special finite representatives. In the \(B,B\) row set \(u=\max(a,c)\), \(v=\min(a,c)\). The induced distinguished label has alpha columns \((P,u)\) and beta columns \((b+d,v)\). Since \(P\geq b+d\geq u\geq v\), its first two orbit columns are \((2P,2(b+d))=(H,X)\). For \(D,B\), its alpha columns are \((P,a+d)\) and its beta columns are \((\max(b,c),\min(b,c))\). Since \(a+d>\max(b,c)\), its first two orbit columns are \((2P,2(a+d)-1)=(H,X)\). The \(B,D\) calculation gives \((2P,2(b+c)-1)=(H,X)\). For \(D,D\), put \(R=a+c\), \(u=\max(b,d)\), \(v=\min(b,d)\). Induction of the special first label and the second two-column label gives alpha columns \((P,u,1)\) and beta columns \((R-1,v)\) when \(u>0\). If \(u=v=0\), the alpha columns are \((P,1)\). Because \(P\geq R\geq b+d+2\), these labels are distinguished. Their first two orbit columns are \((2P,2R-2)=(H,X)\). The remaining columns are \((2u,2v,2)\) if \(v>0\), \((2u,1,1)\) if \(v=00\). By Lemma~\ref{loc:compression}, the source selects the full constant \(E\)-block. Proposition~\ref{ord:containment} therefore assigns this pair to the chosen global orbit; it cannot be assigned to the other very-even orbit, whose closure is incomparable at this partition. When both local factors are nonempty, denote the half labels of the selected pair by \((\epsilon_+,\epsilon_-)\). Lemma~\ref{cl:half-switch} switches both halves simultaneously. This gives the two compatible pairs \[ (\epsilon_+,\epsilon_-),\qquad (-\epsilon_+,-\epsilon_-) \] inside the same global orbit bound. Indeed switching one local half is induced by an isometry of determinant \(-1\) on one summand, so it exchanges the two global orthogonal orbits; switching both has determinant \(+1\) and preserves the global orbit. The two remaining pairs belong to the other fork source. If one factor is empty, the chosen global orbit has only one corresponding local half, already supplied by its fork source. \end{proof} \subsection{Type A and the local conclusion} For a type-\(A\) partition \(\lambda\), take the Young parabolic \[ K_\lambda=\prod_j\mathfrak S_{\lambda'_j}. \] Its longest Schubert variety is a smooth partial flag fiber, so its canonical object is independent of every coefficient characteristic. Its RSK shape is \(\lambda\), and the standard block \(S^\lambda\) occurs by column antisymmetrization. The source is consequently in the exact ordinary cell of \(\lambda\). At a central torus point its coefficient image contains the ordinary column Specht polynomial \(\prod_j\Delta_{\mathfrak S_{\lambda'_j}}\), whose \(W\)-span is the unique minimum copy of \(S^\lambda\). Full-rank stabilizers for the simply connected type-\(A\) torus are central; the ordinary source has trivial cocycle there. This supplies the required minimum copies at all such points. \begin{theorem}\label{cl:classical-local} Let \(G\) be simply connected of classical type and let \(p>h\). At each full-rank torus point \(y\), every local nilpotent orbit \(\mathcal O\) has a finite actual source family whose completed coefficient ideal contains its trivial-local-system minimum Springer copy \(U_{\mathcal O}\), and whose support is contained in the closure of its ambient nilpotent orbit. In even orthogonal type the two very-even closures are treated separately. \end{theorem} \begin{proof} Type \(A\) is the preceding construction, type \(C\) is Proposition~\ref{cl:c-generation}, and odd orthogonal type is Proposition~\ref{cl:b-generation}. For even orthogonal type combine Propositions~\ref{cl:d-generation-nonvery} and \ref{cl:very-even}, using the central homogeneity and half-switch lemmas. The finite parity inputs hold at every odd coefficient characteristic, since the local intersection forms are saturated over \(\mathbb Z[1/2]\). The operator bound in Proposition~\ref{op:full} and the root-sheet construction in Lemma~\ref{cl:root-sheets} require \(p>h\). Explicitly it is \(p>2d\) for \(B_d,C_d\) and \(p>2d-2\) for \(D_d\); it also makes \(p\) prime to the Spin center. \end{proof} For any ambient closed orbit downset \(\mathcal C\), take the finite sums of these source families over its permitted local orbits at each full-rank point. The ordinary inclusion Proposition~\ref{ord:containment} and the downset generation Proposition~\ref{ord:kato} then give the required local equality with \(I_{\mathcal C}\). The general globalization argument applies to these finite source and tensor words. % End sections/classical.tex % Begin sections/exceptional.tex \section{Finite sources in the exceptional types}\label{ex:section} We construct the remaining generators in the finite criterion of Proposition~\ref{loc:finite-criterion}. Throughout this section the nodes of $E_r$ are numbered so that the edges are \[ (1,3),\ (3,4),\ (2,4),\ (4,5),\ldots,(r-1,r). \] A string $i_1\cdots i_d$ denotes the product $s_{i_1}\cdots s_{i_d}$; juxtaposed lines in a displayed string are concatenated. If $J$ is a subset of the finite simple reflections $S$, let $w_J$ be its longest element and $W^J$ the set of minimal representatives of right $W_J$-cosets. All words denoted by $x$ below belong to $W^J$. We use the normalization \[ (H_s+v)(H_s-v^{-1})=0,\qquad C_s=H_s+v. \] In particular, $C_s(1)=1+s$. For $I\subset S$, let $P_I\subset G_{\CC}$ be the standard parabolic subgroup corresponding to $I$. \subsection{Actual elements and finite blocks} An \emph{actual ordinary source} is an element $w$ for which ${}^pC_w=C_w$ in the indicated range of characteristics. This equality identifies an indecomposable parity object, and hence an actual tilting source under the character correspondence used above. Our finite character labels use the ordinary Springer convention: the trivial representation corresponds to the regular orbit. A label $\phi_{d,b}$ denotes an irreducible character of degree $d$ whose first coinvariant occurrence has degree $b$. A representation $F$ with this label is represented in the positive Hecke normalization by \begin{equation}\label{ex:det-convention} F^+=\det\otimes F. \end{equation} An actual source detects $F$ if $C_w(1)$ acts nontrivially on $F^+$. Since the finite source group algebra is semisimple, one nonzero matrix in this block generates its full matrix algebra as a two-sided ideal. The source bound is the affine orbit attached to the ordinary cell of $w$, with the finite-to-affine convention of Proposition~\ref{ord:finite-label}; this bound depends on the special representation of the finite family. We use the following precise finite inputs. The exhaustive calculations in \cite[Appendix, especially \S5.4]{EJ} give ${}^pC_w=C_w$ for every element of every finite Weyl group of rank at most six when $p>7$. In types $F_4$ and $G_2$ the only interesting primes in those calculations are $2$ and $3$, so the corresponding whole-basis equality holds for $p>3$. Here ``interesting'' means that the two bases differ. In particular the $G_2$ case includes characteristic $7$. For a spherical (co)minuscule quotient, the ordinary and $p$-canonical bases agree in good characteristic by \cite[Theorem~9.1]{Baine}. We apply this to the classical minuscule quotients and to $E_7/E_6$. In type $A$, the input needed here is smaller: every ordinary cell has an actual parabolic-longest source. Indeed a permutation which reverses consecutive blocks of sizes $\lambda'_1,\lambda'_2,\ldots$ has Robinson--Schensted shape $\lambda$, and its Schubert variety is a finite flag variety. This gives a source in each cell for every $p$. Here is the finite-algebra reason that these inputs supply the blocks used later. If every maximal right $W_K$-coset element in a finite group $H$ is actual, the cell filtration of $\cH(H)C_{w_K}$ supplies an actual element detecting each ordinary constituent of $\Ind_{W_K}^{H}\det$. To see this, specialize the filtration at $v=1$ and project to a simple matrix block. Its first nonzero cell quotient is the quotient for the family of that constituent; if every basis element of that quotient acted by zero, the quotient would act by zero. Thus some basis element in the required cell has nonzero action. For products of irreducible components, take products of the sources. For a special representation one may use any positive basis element in its cell: the positive special-module theorem gives nonzero action \cite[Propositions~9, 13, 17 and Corollary~16]{KM}. We shall use this last statement only for special representations. The nonspecial actions required beyond the spherical inputs are computed explicitly in Section~\ref{ex:matrices}. In types $E_6,F_4,G_2$, every canonical element of every finite standard affine parabolic is actual, with respective bounds $p>12,p>12,p>6$. For $G_2$ the finite parabolics have types $G_2,A_2,A_1\times A_1$ or smaller, so the same assertion includes $p=7$. Proposition~\ref{ord:finite-generation} expresses each ordinary based downset ideal in terms of these finite elements. Lemma~\ref{cl:matrix-block} therefore puts its entire coefficient image in $I^+_{\cC}$, proving $I^+_{\cC}=I_{\cC}$ in these types. The remaining finite calculation concerns $E_7$ and $E_8$. \subsection{Two propagation lemmas} \begin{lemma}[Partial-Levi inflation]\label{ex:levi} Let $I,J\subset S$, put $K=I\cap J$, and let $x\in W_I$ be minimal in its right $W_J$-coset. If \[ {}^pC_{xw_K}=C_{xw_K}\quad\text{in }\cH(W_I), \] then ${}^pC_{xw_J}=C_{xw_J}$ in $\cH(W)$. \end{lemma} \begin{proof} Let $G_I$ be the standard Levi subgroup of $G_{\CC}$ with nodes $I$. The closed inclusion $G_I/(G_I\cap P_J)\longrightarrow G_{\CC}/P_J$ identifies the Schubert variety indexed by $x$ with the same Schubert variety in the ambient partial flag variety; it identifies their Schubert stratifications as well \cite[Lemma~4.8]{RS}. Normalized smooth pullback along the full-to-partial flag projections takes its indecomposable parity complex to the complexes indexed by $xw_K$ and $xw_J$, respectively. It takes the characteristic-zero intersection cohomology complex to the corresponding normalized pullback. Equality of the parity and intersection cohomology characters therefore transfers between the two full flag varieties. Equivalently, the two relevant spherical canonical coefficients are the coefficients of the same stratified variety. This proves the assertion for an indecomposable source. \end{proof} \begin{lemma}[Common-coset propagation]\label{ex:jensen} Suppose ${}^pC_z=C_z$, $w=sz>z$, and \begin{equation}\label{ex:product} C_sC_z=C_w+\sum_{y\in Y}\mu_y C_y,\qquad \mu_y\in\ZZ_{>0}. \end{equation} For every $y\in Y$, suppose $w,y$ belong to a common left or right coset of a finite standard parabolic $W_I$. In that coset write the upper element as a minimal coset representative times its parabolic factor, using inversion in the left-coset case. Suppose that this parabolic factor $b$ satisfies ${}^pC_b=C_b$. Then ${}^pC_w=C_w$. \end{lemma} \begin{proof} Positivity of the $p$-canonical structure constants and of the transition to the ordinary basis implies \[ {}^pC_w=C_w+\sum_{y\in Y}a_y C_y, \qquad a_y\in\ZZ_{\geq0}[v,v^{-1}]. \] In fact this follows by comparing the ordinary-basis coefficients of the actual product ${}^pC_s\,{}^pC_z=C_sC_z$: the leading summand ${}^pC_w$ occurs with multiplicity one, and all other contributions are nonnegative. Let $h_{y,w}$ and ${}^ph_{y,w}$ be the $H_y$-coefficients of $C_w$ and ${}^pC_w$, respectively. Jensen's parabolic coefficient theorem \cite[Corollary~3.10]{Jensen} and the equality for $b$ give ${}^ph_{y,w}=h_{y,w}$ for each $y\in Y$. Comparing the coefficient of $H_y$ gives \[ 0=\sum_{t\in Y}a_t h_{y,t}. \] All coefficients on the right are nonnegative and $h_{y,y}=1$; hence $a_y=0$. This holds for every $y$. Inversion preserves both canonical bases and proves the left-coset assertion. \end{proof} The hypothesis in Lemma~\ref{ex:jensen} concerns the parabolic upper element. We verify it directly for the two rank-seven upper factors below. All other common-coset factors have connected components of rank at most six. \subsection{The two upper sources for the last geometric construction} We first record these sources explicitly, because both are inputs to the construction in Section~\ref{e8:section}. Let \(J=\{1,3,4,5,6,7,8\}\), of type $A_7$, and define \begin{align*} x_{509}&=\mathtt{43154234565423143542},\\ x_{1221}&=\mathtt{4315436547658765423143542}. \end{align*} Their lengths are $20$ and $25$, respectively. \begin{proposition}\label{ex:inputs} For $p>7$ one has \[ {}^pC_{x_{509}w_J}=C_{x_{509}w_J},\qquad {}^pC_{x_{1221}w_J}=C_{x_{1221}w_J}. \] The first equality uses only the rank-six finite input; the second uses a smooth longest element and rank-two common-coset factors. \end{proposition} \begin{proof} The word $x_{509}$ is supported on $I=\{1,2,3,4,5,6\}$, of type $E_6$, and $I\cap J$ is of type $A_5$. Apply Lemma~\ref{ex:levi} and the rank-six input. For the second equality set $z_0=w_J$. Table~\ref{ex:1221-table} defines $z_k=s_{i_k}z_{k-1}$ for $1\leq k\leq25$, and gives sets $A_k$. The words are reduced and $z_{25}=x_{1221}w_J$. The complete ordinary multiplication identities are \begin{equation}\label{ex:1221-products} C_{s_{i_k}}C_{z_{k-1}} =C_{z_k}+\sum_{j\in A_k}C_{s_jz_{k-1}}. \end{equation} There are $23$ correction terms in total, all with coefficient one. These are identities in the spherical module $\cH(E_8)C_{w_J}$; the multiplication rule (\ref{app:spherical-multiplication}) gives a direct integer Laurent-polynomial verification. More specifically, start with $C_{w_J}$, use the listed $i_k$, and impose bar invariance and strict positive-degree triangularity after each multiplication. The complete list of terms to subtract is exactly the last column of Table~\ref{ex:1221-table}. This is a finite verification with $337$ nonzero spherical terms at the last step. For $j\in A_k$ put $y=s_jz_{k-1}$. The same root-matrix calculation gives $\ell(y)=\ell(z_{k-1})-1$ and \[ z_k=s_{i_k}s_jy. \] Thus $y,z_k$ belong to the same left coset of $W_{\{i_k,j\}}$, a rank-two finite group. Its ordinary and $p$-canonical bases agree for $p>7$. Lemma~\ref{ex:jensen}, starting from the smooth source $w_J$, proves all $25$ assertions in order. \end{proof} % Begin sections/exceptional_1221_steps.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{rccc} \caption{The complete chain for the upper source $x_{1221}w_J$. Here $z_0=w_J$, $z_k=s_{i_k}z_{k-1}$, and the last column is the set $A_k$ in (\ref{ex:1221-products}).}\label{ex:1221-table}\\ \toprule $k$ & $i_k$ & Name of $z_k$ & $A_k$\\\midrule \endfirsthead \toprule $k$ & $i_k$ & Name of $z_k$ & $A_k$\\\midrule \endhead \bottomrule\endfoot 1 & 2 & 2 & $\varnothing$\\ 2 & 4 & 3 & $\{2\}$\\ 3 & 5 & 5 & $\{4\}$\\ 4 & 6 & 8 & $\{5\}$\\ 5 & 7 & 12 & $\{6\}$\\ 6 & 8 & 18 & $\{7\}$\\ 7 & 3 & 26 & $\varnothing$\\ 8 & 4 & 36 & $\{3\}$\\ 9 & 5 & 49 & $\{4\}$\\ 10 & 6 & 66 & $\{5\}$\\ 11 & 7 & 87 & $\{6,8\}$\\ 12 & 2 & 113 & $\varnothing$\\ 13 & 4 & 145 & $\{2\}$\\ 14 & 5 & 183 & $\{4\}$\\ 15 & 6 & 229 & $\{7,5\}$\\ 16 & 3 & 284 & $\varnothing$\\ 17 & 4 & 348 & $\{3\}$\\ 18 & 5 & 422 & $\{6,4\}$\\ 19 & 1 & 507 & $\varnothing$\\ 20 & 3 & 605 & $\{1\}$\\ 21 & 4 & 717 & $\{3\}$\\ 22 & 5 & 834 & $\{4\}$\\ 23 & 6 & 969 & $\{5\}$\\ 24 & 7 & 1101 & $\{6\}$\\ 25 & 8 & 1221 & $\{7\}$\\ \end{longtable} \endgroup % End sections/exceptional_1221_steps.tex \subsection{The additional exceptional elements}\label{ex:elements} Table~\ref{ex:source-words} specifies the nine elements needed in addition to the uniform finite sources. Table~\ref{ex:source-bounds} records the blocks that they detect. The ordinary family label determines the source bound. % Begin sections/exceptional_source_words.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{lll p{8.3cm}} \caption{The nine additional source elements $w=xw_J$. Digit strings are reduced words for $x$.}\label{ex:source-words}\\ \toprule $W$ & Name & $J$ & $x$\\\midrule \endfirsthead \toprule $W$ & Name & $J$ & $x$\\\midrule \endhead \bottomrule\endfoot $E_7$ & $x_{7,43}$ & $\{1,3,4,5,6,7\}$ & $\mathtt{5423143542}$\\ $E_7$ & $x_{7,259}$ & $\{1,3,4,5,6,7\}$ & $\mathtt{31435465765423143542}$\\ $E_8$ & $x_{8,1540}$ & $\{2,3,4,5,6,7\}$ & $\mathtt{3425436542765423145678}$\\ $E_8$ & $x_{8,191}$ & $\{1,2,3,4,5,6,7\}$ & $\begin{gathered}[t]\mathtt{76542345678765423143542654}\\[-2pt]\mathtt{3176542345678}\end{gathered}$\\ $E_8$ & $x_{8,232}$ & $\{1,2,3,4,5,6,7\}$ & $\begin{gathered}[t]\mathtt{24354265431765423456787654}\\[-2pt]\mathtt{231435426543176542345678}\end{gathered}$\\ $E_8$ & $x_{8,2611}$ & $\{1,3,4,5,6,7,8\}$ & $\begin{gathered}[t]\mathtt{24565423457654234587654231}\\[-2pt]\mathtt{43542}\end{gathered}$\\ $E_8$ & $x_{8,418}$ & $\{2,3,4,5,6,7,8\}$ & $\begin{gathered}[t]\mathtt{24315423456542314354276543}\\[-2pt]\mathtt{1}\end{gathered}$\\ $E_8$ & $x_{8,1356}$ & $\{2,3,4,5,6,7,8\}$ & $\begin{gathered}[t]\mathtt{24354265437654287654231435}\\[-2pt]\mathtt{426543765428765431}\end{gathered}$\\ $E_8$ & $x_{8,2794}$ & $\{1,3,4,5,6,7,8\}$ & $\begin{gathered}[t]\mathtt{31435465431765423458765423}\\[-2pt]\mathtt{143542}\end{gathered}$\\ \end{longtable} \endgroup % End sections/exceptional_source_words.tex \begin{table}[ht] \centering\small \begin{tabular}{lllrl} \toprule Type & Source & Ordinary block $F$ & $a$ & Bound\\ \midrule $E_7$ & $x_{7,43}w_{A_6}$ & $\phi_{189,22}$ &22&$A_2+2A_1$\\ $E_7$ & $x_{7,259}w_{A_6}$ & $\phi_{15,28}$ &25&$A_2+A_1$\\ $E_8$ & $x_{8,1540}w_{D_6}$ & $\phi_{1400,32}$ &32&$D_4(a_1)+A_1$\\ $E_8$ & $x_{8,191}w_{E_7}$ & $\phi_{35,74}$ &74&$2A_1$\\ $E_8$ & $x_{8,232}w_{E_7}$ & $\phi_{8,91}$ &91&$A_1$\\ $E_8$ & $x_{8,2611}w_{A_7}$ & $\phi_{1050,34}$ &32&$D_4(a_1)+A_1$\\ $E_8$ & $x_{8,418}w_{D_7}$ & $\phi_{560,47}$ &47&$A_2+2A_1$\\ $E_8$ & $x_{8,1356}w_{D_7}$ & $\phi_{210,52},\phi_{50,56}$ &52&$A_2+A_1$\\ $E_8$ & $x_{8,2794}w_{A_7}$ & $\phi_{175,36}$ &32&$D_4(a_1)+A_1$\\ \bottomrule \end{tabular} \caption{Additional detected blocks and their source bounds.} \label{ex:source-bounds} \end{table} \begin{proposition}\label{ex:actual} The first eight elements in Table~\ref{ex:source-bounds} are actual for $p>7$. The last element is actual for $p>30$. They detect the listed blocks and have the listed source bounds. \end{proposition} \begin{proof} The short word for $x_{7,43}$ is supported on $I=\{1,2,3,4,5\}$, of type $D_5$. Here $I\cap J$ has type $A_4$. Lemma~\ref{ex:levi} proves its actuality directly from the finite rank-five input. All the other assertions of actuality in the first eight rows are proved by the complete common-coset tables in Appendix~\ref{app:dag}. We describe the roots of these arguments to make their dependencies explicit. The $E_7/A_6$ chain for $259$ starts at the word $\mathtt{2315436542}$, supported on $E_6$. The $E_8/D_6$ chain for $1540$ starts at $\mathtt{145678}$, whose support is $A_1+A_5$. The $E_8/E_7$ chains for $191$ and $232$ start at $\mathtt{345678}$, with support $A_6$. The $E_8/A_7$ chain for $2611$ starts at $\mathtt{65423143542}$, supported on $E_6$. Each seed is actual by Lemma~\ref{ex:levi}. Every ordinary product, lower correction, and common-coset relative word is printed in the appendix, so induction with Lemma~\ref{ex:jensen} proves the desired equalities. There are three auxiliary $E_7/D_5$ elements, named $1104,1455,1486$. Their words and complete chains are also given in Appendix~\ref{app:dag}. Their common seed is supported on $\{2,3,4,5,6,7\}$, of rank six, and every subsequent correction factor has rank at most six. The $E_8/D_7$ element $418$ is the partial-Levi inflation of $1486$. The chain for $1356$ starts with the inflation of $1455$ and uses $1104$ once as a common-coset upper factor. In these two inflations $I=\{1,\ldots,7\}$ and $I\cap J=\{2,\ldots,7\}$ is $D_6$; the equality of the full upper element with the certified $E_7/D_5$ word is checked as an equality of Weyl-group elements. The notation $E_7/D_5$ for the auxiliary construction records the quotient used to construct that word; the inflations use the intersection $D_6$ computed here. The remaining rank-seven upper factor, in the chain for $2611$, is the $E_7/A_6$ element with short word $\mathtt{4315436542}$. Its short word is supported on $E_6$, so it is itself actual by Lemma~\ref{ex:levi}. These observations close every leaf of the common-coset construction in a finite rank-at-most-six input. Finally the last row is Theorem~\ref{e8:source}, using Proposition~\ref{ex:inputs} for its two upper sources. For ordinary cell membership use the complete star paths in Table~\ref{app:star-paths}. They take each displayed full word to the representative cell of \cite[\S6]{GH} identified in Table~\ref{app:star-endpoints}. Every step is specified by an adjacent pair and is verified by the integer root matrices \eqref{app:root-matrices}. Left and right star operations preserve two-sided cells. In the order of Table~\ref{ex:source-bounds}, the resulting special characters are \[ \begin{array}{c|l} E_7&\phi_{189,22},\ \phi_{120,25},\\ E_8&\phi_{1400,32},\ \phi_{35,74},\ \phi_{8,91},\ \phi_{1400,32},\ \phi_{560,47},\ \phi_{210,52},\ \phi_{1400,32}. \end{array} \] Their minimum degrees give the printed $a$-values. Proposition~\ref{ord:finite-label}, with the Springer tables in \cite{LusztigIV,GeckPfeiffer}, gives the affine bounds. The positive special-module theorem proves detection for the special blocks. The four additional nonspecial blocks are verified by the explicit actions in Section~\ref{ex:matrices}. This proves all the assertions, with the determinant conversion (\ref{ex:det-convention}). \end{proof} There is one further source inside the affine $D_8$ subgroup of $E_8$. It is the pure positive $(5,3)$ source of the classical construction. It detects the $D_8$ character $\texttt{.22211}$, whose family has $a=24$, with ambient bound $A_4+2A_1$. This source is recorded as mask $510$, row $28$ in Table~\ref{app:blocks-8}. Its actuality is the classical geometric construction, while its finite family and embedding determine the displayed bound. \subsection{The nonspecial matrix actions}\label{ex:matrices} The following exact actions prove detection of the four nonspecial blocks used here. Let $R$ be an integral representation of the finite Weyl group, set \[ e_J=\sum_{u\in W_J}R(u),\qquad C_{xw_J}=\sum_{y\in W^J}q_y(v)H_yC_{w_J}. \] Then its specialization acts as \begin{equation}\label{ex:action-formula} Z_x=\left(\sum_yq_y(1)R(y)\right)e_J. \end{equation} The integral simple-reflection matrices, their class character checks, and the vectors used in (\ref{ex:action-formula}) form part of the explicit certificates in Appendix~\ref{app:matrices}. The auxiliary prime used in these checks identifies the integral representations in characteristic zero. For $E_7$, take the $15$-dimensional representation $\phi_{15,7}=\det\otimes\phi_{15,28}$. Its simple-reflection traces are $5$, its commutant has dimension one, and $\operatorname{tr}(e_J)/5040=1$ for $J=A_6$. The ordinary spherical vector for $x_{7,259}$ has $141$ nonzero terms. Direct substitution into (\ref{ex:action-formula}) gives \begin{equation}\label{ex:e7-action} \operatorname{tr}(Z_{259})=20160,\qquad \sum_{a,b}(Z_{259})_{ab}^{\,2}=2649894912. \end{equation} In particular this actual source detects the nonspecial ordinary block $\phi_{15,28}$, with bound $A_2+A_1$. For each of the three $E_8$ representations of dimension $d$, set \[ (v_0)_i=((37i+11)\bmod101)-50\quad(1\leq i\leq d), \qquad u=e_Jv_0, \] where the remainder lies in $\{0,\ldots,100\}$. Then $u$ is integral and $W_J$-invariant. The exact nonzero vectors $z=\sum_yq_y(1)R(y)u$ are as follows. \begin{table}[ht] \centering\small \begin{tabular}{llrr} \toprule Source & Positive representation & $z_1$ & $\sum_i z_i^2$\\ \midrule $2611$ & $\phi_{1050,10}$ & $28800$ & $1839562447896576$\\ $1356$ & $\phi_{50,8}$ & $5160960$ & $1165436657860608$\\ $2794$ & $\phi_{175,12}$ & $-241920$ & $17245228105728$\\ \bottomrule \end{tabular} \caption{Exact integral nonzero actions in the three nonspecial $E_8$ blocks.} \label{ex:e8-actions} \end{table} The simple-reflection matrices satisfy the Coxeter relations, and their traces on all $112$ conjugacy classes identify the indicated characters. Integer finite-order traces have absolute value at most the dimension. Thus the class checks modulo $1000003$, which is larger than twice all three dimensions, determine the integer traces uniquely. The ordinary spherical coefficients in the action formula are obtained from bar invariance and triangularity; there are $897$ nonzero terms for $2794$. Equation~(\ref{ex:action-formula}) and Table~\ref{ex:e8-actions} prove nonzero action. Actuality of its last row comes separately from Theorem~\ref{e8:source}. \subsection{The two local constituents detected by the last source} \label{ex:last-restrictions} The two remaining local representations can be verified directly in the same $175$-dimensional representation. This also records the multiplicities needed in the final coverage argument. Write $s_0=s_\theta$, where \[ \theta=2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4 +5\alpha_5+4\alpha_6+3\alpha_7+2\alpha_8. \] The following explicit conjugation expresses its matrix in terms of the eight simple generators: \begin{equation}\label{ex:highest-reflection} s_0=u s_8u^{-1},\qquad u=\mathtt{8765423143542654317654234567}. \end{equation} The word for $u$ has length $28$, so this is a $57$-letter reflection word. The equality follows either by applying it to the eight simple roots, or by verifying $u(\alpha_8)=\theta$. \begin{lemma}\label{ex:last-two} In the ordinary Springer convention, \begin{align*} [\Res_{W(E_6)\times W(A_2)}^{W(E_8)}\phi_{175,36}: \phi_{10,9}\boxtimes\mathbf1]&=1,\\ [\Res_{W(A_5)\times W(A_2)\times W(A_1)}^{W(E_8)}\phi_{175,36}: S^{(3,3)}\boxtimes\mathbf1\boxtimes\mathbf1]&=1. \end{align*} The first subgroup is obtained by deleting node $7$ from the extended diagram, and the second by deleting node $4$. Both local representations have ambient orbit $D_4(a_1)+A_1$. \end{lemma} \begin{proof} Use the positive representation $V=\phi_{175,12}$. For the first subgroup, $E_6$ has nodes $1,\ldots,6$ and $A_2$ has nodes $8,0$. The sign projector \[ P_{A_2}^-=\frac{1}{6}\sum_{a\in W(A_2)}\det(a)R(a) \] has image of dimension $10$. The six $E_6$ generators commute with this projector, and their action on its image has a one-dimensional commutant. Thus the image is the unique degree-$10$ irreducible of $E_6$, namely $\phi_{10,9}$, with multiplicity one. Determinant twist makes the $A_2$ factor trivial, and $\phi_{10,9}$ is self-associated under determinant twist. For the second subgroup the factors have nodes \[ \{5,6,7,8,0\},\qquad\{1,3\},\qquad\{2\}, \] respectively. The product of the $A_2$ and $A_1$ sign projectors has image of dimension $5$ and the induced $A_5$ action has one-dimensional commutant. In the order of cycle types \[ 6,\ 51,\ 42,\ 411,\ 33,\ 321,\ 3111,\ 222,\ 2211,\ 21111,\ 111111 \] its character is \[ 0,\ 0,\ -1,\ 1,\ 2,\ -1,\ -1,\ 3,\ 1,\ -1,\ 5. \] The Murnaghan--Nakayama rule identifies this with $S^{(2,2,2)}$. Determinant twist gives $S^{(3,3)}$ and makes both smaller factors trivial. For completeness, the dimensions and commutant assertions are exact rational statements. The matrices of the projectors are rational with denominators dividing $12$. Their images and commutants can be computed after reduction modulo $1000003$. The computed image dimensions agree with the projector traces, hence with the rational ranks. A commutant of dimension one over this finite field implies rational commutant dimension at most one, since rank cannot increase upon reduction. Scalar matrices give the reverse inequality; semisimplicity over $\CC$ then gives irreducibility. These are ordinary characteristic-zero restriction calculations, independent of the characteristic $p$ of the theorem. The embedded weighted Dynkin diagrams of their Springer orbits give $D_4(a_1)+A_1$ in both cases. They appear explicitly in Tables~\ref{app:witness-8-4} and~\ref{app:witness-8-9}. \end{proof} \subsection{Exact local trace and support cuts}\label{ex:trace-cuts} We describe the finite calculation underlying the coverage tables. This description includes the phases at noncentral points. Let $y$ be a full-rank point of the coefficient torus, $M=W_y$, and $E\in\Irr(M)$. Let $H$ be the projection of a finite affine standard parabolic, with rational fixed weight $\xi$. Its elements are $t_{\xi-h\xi}h$, $h\in H$. For $z\in Wy$, choose $r_z$ with $r_z z=y$. The trace on the fibre induced from $E$ is \begin{equation}\label{ex:phase-trace} \chi_{E,y,\xi}(h)= \sum_{\substack{z\in Wy\\ hz=z}} z^{\,\xi-h\xi}\, \chi_E(r_zhr_z^{-1}). \end{equation} The difference $\xi-h\xi$ is integral, so the phase is a well-defined root of unity. The value is independent of the chosen $r_z$. For a detected ordinary block $F$ define \begin{equation}\label{ex:trace-multiplicity} m(F,E;y,\xi) =\frac{1}{|H|}\sum_{h\in H} \overline{\chi_F(h)}\,\chi_{E,y,\xi}(h). \end{equation} This is a nonnegative integer. All finite characters in this formula use the ordinary convention; the determinant in (\ref{ex:det-convention}) has already been incorporated in passing from a positive Hecke source to its ordinary block. The fixed-point sum uses the entire intersection $H\cap W_z$. When $\xi$ is integral, conjugation by the global monomial of weight $\xi$ removes the cocycle and the matrix of multiplicities is \[ (\Ind_H^W)^{\mathsf t}\Ind_M^W. \] In general (\ref{ex:phase-trace}) is evaluated in $\ZZ[\zeta_e]$, where $e$ is the order of $y$ and $\zeta_e$ is a primitive $e$-th root of unity. For every $E$ one also has the degree identity \begin{equation}\label{ex:degree-check} \sum_{F\in\Irr(H)}\dim(F)m(F,E;y,\xi) =[W:M]\dim(E). \end{equation} For each ordinary parabolic subgroup $L$ of $M$, let $D_L$ be the ambient orbit attached to $j_L^W\det_L$. A cut associated with $L$ retains $E$ whenever \[ [\Res_L^M E:\det_L]>0. \] By Corollary~\ref{op:retention} the corresponding allowed tensor operations retain the required generator and impose the upper bound $\overline{D_L}$. Define the finite orbit down-set \begin{equation}\label{ex:allowed-cut-set} \mathcal D(E)= \bigcap_{\substack{L\text{ ordinary parabolic in }M\\ [\Res_L^M E:\det_L]>0}} \overline{D_L}. \end{equation} Thus each cut is indexed by an ordinary parabolic of the local centralizer $M$. Thus an actual source block $F$ with bound $B_F$ supplies the local generator attached to $E$ if \begin{equation}\label{ex:finite-test} m(F,E;y,\xi)>0,\qquad \overline{B_F}\cap\mathcal D(E) \subseteq\overline{\cO(E)}. \end{equation} Here $\cO(E)=G_{\CC}\cdot\cO_M(E)$ is the ambient nilpotent orbit containing the local Springer orbit. It is computed by embedding the local weighted Dynkin diagram and taking its ambient dominant representative. The constant matrix-block test in the proof of Proposition~\ref{ord:containment} also gives $\cO(E)\subseteq\overline{B_F}$ whenever the displayed multiplicity is positive. \subsection{Coverage of the exceptional local generators} The full-rank centralizers and the numbers of required generators are recorded in Table~\ref{ex:vertex-counts}. The order of $y$ is its order in the torus of the simply connected datum. Central translates are retained for $E_7$. \begin{table}[ht] \centering\small \begin{tabular}{crlr@{\qquad}crlr} \toprule \multicolumn{4}{c}{$E_7$}&\multicolumn{4}{c}{$E_8$}\\ Vertex&Order&Centralizer&Rows&Vertex&Order&Centralizer&Rows\\ \midrule 1&1&$E_7$&45&1&1&$E_8$&70\\ 2&2&$E_7$&45&2&2&$D_8$&75\\ 3&2&$D_6+A_1$&62&3&2&$E_7+A_1$&90\\ 4&2&$D_6+A_1$&62&4&3&$E_6+A_2$&63\\ 5&3&$A_5+A_2$&33&5&3&$A_8$&30\\ 6&4&$2A_3+A_1$&50&6&4&$D_5+A_3$&80\\ 7&4&$A_7$&22&7&4&$A_7+A_1$&44\\ 8&6&$A_5+A_2$&33&8&5&$2A_4$&49\\ &&&&9&6&$A_5+A_2+A_1$&66\\ \midrule \multicolumn{3}{r}{Total}&352&\multicolumn{3}{r}{Total}&567\\ \bottomrule \end{tabular} \caption{The full-rank local generators with trivial local system.} \label{ex:vertex-counts} \end{table} \begin{proposition}[Exceptional finite coverage]\label{ex:coverage} For $E_7$ and $p>18$, respectively for $E_8$ and $p>30$, every local generator required by Proposition~\ref{loc:finite-criterion} satisfies (\ref{ex:finite-test}) for an actual source. There are $352$ such full-rank rows in $E_7$ and $567$ in $E_8$. \end{proposition} \begin{proof} The selected finite blocks are specified, including their affine node subsets, ordinary character labels and support bounds, in Tables~\ref{app:blocks-7} and~\ref{app:blocks-8}. Their sources are the finite rank and minuscule inputs, the type-$A$ longest elements, Proposition~\ref{ex:actual}, and the single classical $D_8$ source described above. The bounds are computed by taking the special representation of each finite family, applying truncated induction to the ambient finite Weyl group, and using its ordinary affine-cell orbit. The defining truncated constituent has multiplicity one and matching $b$-value. Tables~\ref{app:witness-7-1}--\ref{app:witness-7-8} and \ref{app:witness-8-1}--\ref{app:witness-8-9} give one source for every required row. For a row, use its local orbit with trivial local system to determine $E$ by Springer correspondence. Evaluate (\ref{ex:phase-trace}) and (\ref{ex:trace-multiplicity}) for the specified mask and block. Its multiplicity is positive. For all ordinary parabolic $L\subset M$, compute the sign-restriction multiplicity and retain exactly those $L$ for which it is positive. Intersect the corresponding orbit down-sets with the recorded source down-set. The result is contained in the target down-set printed in the same row. Appendix~\ref{app:finite} gives the complete deterministic rules for these finite calculations; the attached certificates supply integral representation and character data, exact algorithms for the cyclotomic traces, and orbit incidence data. For the two rows using source $8.\mathrm G$, Lemma~\ref{ex:last-two} gives multiplicity one, and the source bound equals the ambient target. All remaining rows are evaluated by the same trace-and-cut rule. The lists comprise every trivial-local-system Springer pair in each centralizer in Table~\ref{ex:vertex-counts}; hence the numbers $352$ and $567$ count all the required generators. The finite criterion now supplies the corresponding completed Springer ideals. The specialization and globalization result of Theorem~\ref{loc:globalization} passes from these full-rank vertices to all semisimple strata. The trivial-local-system generators suffice by the ordinary generation theorem of Proposition~\ref{ord:kato}. \end{proof} Together with the classical constructions, this proves the source condition in every irreducible root system. Tensor products of sources give the same condition for a product of root systems. % End sections/exceptional.tex % Begin sections/e8_geometry.tex \section{Integral geometry of the final \texorpdfstring{\(E_8\)}{E8} source} \label{e8:section} We give an integral geometric construction of the remaining source. All varieties in this section are complex varieties. A prime appearing as a coefficient characteristic is the prime of the parity category, and ultimately is the characteristic \(p>30\) of the reductive group. We use Chow degree, so that a class in \(H^{2d}\) has degree \(d\). For a vector bundle \(E\), write \(e(E)=c_{\operatorname{rank}E}(E)\) for its top Chern class. For a variety of complex dimension \(d\), an IC exponent \(r\) at a point denotes the stalk group in degree \(-d+2r\). When these are the only stalk degrees, the IC polynomial records their ranks as the coefficients of \(q^r\), with \(q\) a formal variable. \subsection{The Schubert varieties and their words} Use the Bourbaki diagram with edges \[ 1\!-\!3\!-\!4\!-\!5\!-\!6\!-\!7\!-\!8, \qquad 2\!-\!4, \] and put \(J=\{1,3,4,5,6,7,8\}\). The Levi factor of \(P_J\) has derived group \(\SL_8\). If \(u\) is a minimal right \(W_J\)-coset representative, write \(X_u\subset G_{\CC}/P_J\) for its Schubert variety and \(\mathbf c_u\) for its spherical canonical element. Normalized smooth pullback to the full flag variety sends \(\mathbf c_u\) to \(C_{u w_J}\). The elements used below are defined by the following reduced words; the entries are multiplied in their displayed order: \begin{align} a={}&(4,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,2),\label{e8:aword}\\ b={}&(6,5,4,3,1,7,6,5,4,3,8,7,6,5,4,2),\label{e8:bword}\\ c={}&(4,3,1,5,4,3,6,5,4,7,6,5,8,7,6,5,4,2,3,1,4,3,5,4,2), \label{e8:cword}\\ z={}&(3,1,4,3,5,4,6,5,4,3,1,7,6,5,4,2,\notag\\[-2pt] &\hspace{24mm}3,4,5,8,7,6,5,4,2,3,1,4,3,5,4,2). \label{e8:zword} \end{align} Thus their partial dimensions are \(20,16,25,32\), respectively. The source called \(2794\) in the finite tables is \(z w_J\). The numbers \(509,277,1221\) refer to \(a w_J,b w_J,c w_J\). The displayed words specify these objects. \begin{theorem}\label{e8:source} For every prime \(p>30\), \[ {}^p C_{z w_J}=C_{z w_J}. \] \end{theorem} The inputs from Proposition~\ref{ex:inputs} are the \(p\)-independence of \(a w_J\) and \(c w_J\). We prove the needed statement for \(b w_J\) below. The proof of Theorem~\ref{e8:source} occupies the rest of this section. \subsection{The decomposable cone} Let \(V\) be an eight-dimensional vector space. We use the finite central covering \(\GL(V)\to L\) of the Levi for which the first graded representation is \(\bigwedge^3V\). Its kernel consists of scalar cube roots of unity and acts trivially on \(\bigwedge^3V\) and \(\bigwedge^6V\). Thus the varieties and incidence maps below descend to \(L\). The opposite big cell at the base point of \(G_{\CC}/P_J\) is the negative unipotent radical. Exponential coordinates are equivariant for the Levi, and its first two graded pieces are \[ \bigwedge\nolimits^3 V,\qquad \bigwedge\nolimits^6 V. \] The third graded piece has dimension eight. The constructions below have zero coordinate in that piece. The opposite chart of \(X_{s_2}\cong\mathbf P^1\) is the root line of \(-\alpha_2\). In \(\bigwedge^3V\), it is a decomposable line. Its Levi saturation is therefore \[ \{\,v_1\wedge v_2\wedge v_3:v_i\in V\,\}, \] the affine cone over \(\Gr(3,V)\). This Levi saturation is also the relevant parabolic saturation. Indeed, for a positive root \(\gamma\ne\alpha_2\), the simply laced root string shows that conjugating its root group by \(\exp(t f_{\alpha_2})\) keeps it in \(P_J\). It fixes the affine root line. The root group of \(\alpha_2\) preserves the whole projective line. Thus the radical of \(P_J\) preserves \(X_{s_2}\), and its Levi saturation has \(P_J\)-stable closure. For the Weyl-group identification, begin with \(s_2w_J\) and apply left Demazure multiplication successively at \[ (4,3,1,5,4,3,6,5,4,7,6,5,8,7,6). \] Each indicated multiplication increases length, and the result is \(b w_J\), of length \(44\). To verify the assertion, use \[ s_i(\alpha_j)=\alpha_j-a_{ij}\alpha_i \] for the \(E_8\) Cartan matrix and count positive roots sent negative. These formulas also verify the word equality. It follows that the displayed cone is the opposite chart of \(X_b\). The nonzero cone is a single Levi orbit, and the \(\alpha_2\)-subgroup moves the other point of the root projective line into its nonzero affine part. Hence the projective variety has only the smooth nonvertex orbit and the vertex for the purpose of the local calculation. We record the integral calculation for this cone, and also for a smaller cone used later. For \(\Gr(3,n)\), let \(\mathcal P_d(3,n-3)\) be the partitions of \(d\) in the \(3\) by \(n-3\) rectangle. In the Schubert basis, multiplication by the positive Plucker class has matrix \begin{equation}\label{e8:pieri} P_d(\mu,\lambda)= \begin{cases} 1,&\mu/\lambda\text{ consists of one box},\\ 0,&\text{otherwise}, \end{cases} \quad \lambda\in\mathcal P_d(3,n-3),\ \mu\in\mathcal P_{d+1}(3,n-3). \end{equation} For \(n=8\), the ranks by degree are \[ 1,1,2,3,4,5,6,6,6,6,5,4,3,2,1,1. \] The nonzero Smith factors of every \(P_d\) are \(1\), with the following exceptions: \[ \begin{array}{c|c} d&\text{nonzero Smith factors}\\ \hline 6&(1,1,1,1,1,3)\\ 7&(1,1,1,1,1,2)\\ 8&(1,1,1,1,1,3). \end{array} \] These small integer matrices are determined completely by \eqref{e8:pieri}; row and column elimination over \(\ZZ\) gives the displayed diagonal forms. In particular every kernel and cokernel is free after inverting \(6\). The punctured affine cone is the complement of the zero section in \(\mathcal O_{\Gr}(-1)\). Its Gysin sequence identifies the relevant link cohomology with the kernels and cokernels of the adjacent Plucker maps. Truncation at the middle degree gives the stalk of the cone IC, and Verdier duality gives its costalk. It follows that the IC over \(\ZZ[1/6]\) has free even stalk and costalk, compatible with reduction modulo every prime greater than \(3\). Thus \begin{equation}\label{e8:b-independent} {}^p C_{b w_J}=C_{b w_J}\qquad(p>3). \end{equation} The Gysin sequence uses precisely the adjacent maps in \eqref{e8:pieri}. \subsection{The contact double cover and its induction} Fix a six-plane \(W\subset V\). The subgroup on nodes \(\{1,\ldots,6\}\) is of type \(E_6\); its partial flag variety at node \(2\) is the adjoint contact variety, of dimension \(21\). Its Schubert divisor has word \(a\) and dimension \(20\). Here is the equation of its opposite chart. For \(\omega\in\bigwedge^3 W\), define \[ K_\omega:W^*\longrightarrow W^*\otimes\det W,\qquad K_\omega(\xi)=(\iota_\xi\omega)\wedge\omega, \] using \(\bigwedge^5W\cong W^*\otimes\det W\). The invariant quartic \(Q_W(\omega)\in(\det W)^2\) is, up to a fixed nonzero scalar, \(\frac16\operatorname{tr} (K_\omega^2)\). With a compatible normalization, the chart of the divisor is \begin{equation}\label{e8:contact} Y_W=\{(\omega,\eta)\in\bigwedge\nolimits^3W\oplus\det W: \eta^2+Q_W(\omega)=0\}. \end{equation} This equation also follows directly from the contact grading. Let \(\beta\) be the highest root of this \(E_6\), and choose root vectors \(e_\beta,f_\beta\) in its \(\mathfrak{sl}_2\)-triple. In the lowest-root coordinate of \(\exp(\operatorname{ad}(\omega+\eta f_\beta))e_\beta\), the possible terms have degrees \(\eta^2,\eta\omega^2\), and \(\omega^4\). The first is nonzero by the root \(\mathfrak{sl}_2\), the middle term vanishes because the exterior-cube pairing is alternating, and the last is the nonzero contact quartic. The latter contact identity is also given in \cite[\S1.2]{Abuaf}. The lowest-root coordinate defines a Borel-stable hyperplane section. The adjoint highest weight is the fundamental weight at node \(2\), so this hyperplane section represents the positive generator of the Picard group. It is therefore the unique Schubert divisor, with multiplicity one. This proves \eqref{e8:contact}. Rescaling its nonzero complex coefficients gives an isomorphic complex variety. Let \(Q_L\) be the parabolic in the Levi \(L\) stabilizing \(W\). The equation and its projective closure are \(Q_L\)-stable. The associated variety is projective over \(L/Q_L=\Gr(6,V)\), with projective fiber \(X_a\). Consequently it has a proper morphism \begin{equation}\label{e8:proper} \pi:L\times_{Q_L}X_a\longrightarrow G_{\CC}/P_J. \end{equation} On the opposite chart it is the closed incidence projection \[ \left\{\begin{array}{l} W\in\Gr(6,V),\quad \omega\in\bigwedge^3W,\quad\eta\in\det W,\\ \eta^2+Q_W(\omega)=0 \end{array}\right\} \longrightarrow\bigwedge^3V\oplus\bigwedge^6V. \] A trivector of support six determines \(W\) uniquely, and that support is recovered regularly on opens defined by nonzero support minors. Thus \(\pi\) is an isomorphism on its support-six locus, including the quartic branch. It is birational onto its image. Let \(U\) be the unipotent radical of \(P_J\). The morphism is \(P_J\)-equivariant. To check the radical, consider a positive root outside the indicated \(E_6\). Its expansion retains a positive coefficient at node \(7\) or \(8\) under conjugation by the \(E_6\) subgroup; every resulting root is positive. This root group fixes the embedded \(E_6\) contact variety pointwise. The remaining radical root groups lie in the \(E_6\) Borel and preserve its Schubert divisor. This gives the action on \(P_J\times_{U\rtimes Q_L}X_a\), whose underlying associated variety is the domain of \eqref{e8:proper}. Its image is the Schubert variety \(X_z\). Explicitly, begin with \(a w_J\), of length \(48\), and apply left Demazure multiplication in the order \begin{equation}\label{e8:demazure} (7,6,5,4,3,1,8,7,6,5,4,3). \end{equation} All twelve steps increase length. The resulting root matrix is that of \(z w_J\), of length \(60\), by the simple-reflection formula above. Parabolic saturation therefore identifies the reduced image with \(X_z\). The geometry is a double cover of the support-six trivector locus. For a generic \(\omega=a_1+a_2\), with \(a_i\) volume forms of complementary three-spaces, the two choices of \(\eta\) are proportional to \(\pm a_1\wedge a_2\). Interchanging the three-spaces interchanges the sheets. This monodromy determines the Fourier summand below. Fix a basis \(e_1,\ldots,e_8\) of \(V\) and write \(e_{ijk}=e_i\wedge e_j\wedge e_k\). The orbit types needed for the fibers are as follows: \[ \begin{array}{c|c|c|c} \text{support}&\text{representative or condition} &\text{partial orbit dimension}&\pi^{-1}(\omega,\eta)\\ \hline 6&Q_W(\omega)\ne0&32&\mathrm{pt}\\ 6&Q_W(\omega)=0,\ \omega\text{ of support }6&31&\mathrm{pt}\\ 5&e_{123}+e_{145}&25&\mathbf P^2\\ 3&e_{123}&16&\Gr(3,5)\\ 0&0&0&\Gr(6,8). \end{array} \] The two support-six entries are understood in the cover. The classification follows from the normal forms for trivectors supported on a six-space; see \cite[Table~I]{Djokovic}. For support at most five, \(\eta=0\), and the fiber consists of six-planes containing the support. This proves the last column directly. These are all the positive-dimensional fiber types. The three lower stabilizers are connected. In support three, their reductive part is \(\SL_3\times\GL_5\); in support five it is \(\operatorname{GSp}_4\times\GL_3\). The remaining factors are block-triangular unipotent groups, and the scalar on the distinguished line in support five is fixed by the symplectic similitude. Thus every irreducible equivariant local system on these lower strata is trivial. These chart computations cover the projective target. Every Schubert closure contains the base point, and the opposite chart is an open neighborhood of that point. Its intersection with a Schubert closure is a nonempty open subset, hence meets the dense Schubert cell. Since the Borel is contained in \(P_J\), every \(P_J\)-orbit meets the chart. Equivariance therefore transports each local computation to the entire corresponding projective orbit. \subsection{The characteristic-zero decomposition} Let \(K=\{1,3,4,5,6,8\}\subset J\), of type \(A_5+A_1\). There are \(28\) minimal right cosets \(W_J/W_K\), and their largest length is \(12\). Associated induction has the spherical character \begin{equation}\label{e8:induction} \left(\sum_{u\in W_J^K}v^{12-\ell(u)}H_u\right)\mathbf c_a =\mathbf c_z+(v+v^{-1})\mathbf c_c +(v^2+1+v^{-2})\mathbf c_b +(v^2+1+v^{-2})\mathbf c_1 . \end{equation} Here \(1\) denotes the zero-dimensional partial Schubert variety, and \(W_J^K\) consists of the elements with no right descent in \(K\). Formula \eqref{e8:induction} is a finite canonical-basis identity. Enumerate those \(28\) cosets using the root matrices. In the ambient spherical module, put \(m_x=H_xC_{w_J}\) for \(x\in W^J\). The ordinary Hecke rule is \[ H_s m_x= \begin{cases} m_{sx},&\ell(sx)>\ell(x),\ sx\in W^J,\\ m_{sx}+(v^{-1}-v)m_x,&\ell(sx)<\ell(x),\\ v^{-1}m_x,&sxW_J=xW_J. \end{cases} \] Construct \(\mathbf c_a\) by bar-triangular recursion, apply the \(28\) operators \(H_u\) in \eqref{e8:induction}, and eliminate leading terms using the unique bar-invariant vectors \(\mathbf c_x\in m_x+\sum_{x'3\). At the vertex the same statement follows from \eqref{e8:originsmith}. These computations cover every projective orbit by the equivariance argument following \eqref{e8:proper}. The lower indecomposable IC objects are also \(p\)-independent: \(X_c\) by Proposition~\ref{ex:inputs}, \(X_b\) by \eqref{e8:b-independent}, and the point trivially. For a parity complex \(\mathscr F\) and the inclusion \(i\) of a closed stratum, the ranks of the canonical maps \(i^!\mathscr F\to i^*\mathscr F\) determine the summands supported there by \cite[Proposition~2.26]{JMW}. Apply this criterion to the parity direct image along the dense Schubert cell of each lower support, after removing the boundary of that cell. The cell is then closed and contractible, so its star and shriek restrictions are direct sums of shifted constant local systems. Restriction to this open union of Schubert strata preserves nonzero indecomposable parity complexes by \cite[Proposition~2.11]{JMW}; hence these local multiplicities are the global multiplicities. All its ranks are the characteristic-zero ranks. Therefore the lower direct summands occur with the multiplicities in \eqref{e8:induction} over every field of characteristic \(p>30\). The remaining top indecomposable parity summand has character \(\mathbf c_z\). Normalized smooth pullback to the full flag variety gives \({}^pC_{zw_J}=C_{zw_J}\), as asserted. \end{proof} % End sections/e8_geometry.tex % Begin sections/conclusion.tex \section{The support formula for all highest weights} \label{sec:conclusion} \subsection{Equality of character spaces} We first assume that the root system is irreducible and that \(G\) is simply connected. Theorem~\ref{cl:classical-local} provides the finite local generators in every classical type. For types \(E_7,E_8\) they are provided by Proposition~\ref{ex:coverage}, with the last geometric source supplied by Theorem~\ref{e8:source}. The finite criterion includes every full-rank stabilizer and every trivial-local-system Springer generator at that stabilizer. In particular the exceptional argument uses only the characteristic bounds stated there. For these types, apply Theorem~\ref{loc:globalization} to each orbit downset \(\cC\). In types \(E_6,F_4,G_2\), the finite generation argument at the start of Section~\ref{ex:section} gives the same conclusion, including \(p=7\) for \(G_2\). Thus in every type, \[ I^+_{\cC}=I_{\cC}. \] The coefficient correspondence of Proposition~\ref{op:coefficient} now gives the equality \begin{equation}\label{end:characters} J^+_{\cC}=J^q_{\cC}. \end{equation} \subsection{Recovering the original label} Let \(\lambda\) be a regular dominant weight and put \(\cC=\overline{\cO_\lambda}\). The quantum support theorem places \(\ch T_q(\lambda)\) in \(J^q_{\cC}\). By \eqref{end:characters}, there is a finite expression \begin{equation}\label{end:triangular} \ch T_q(\lambda)=\sum_{\mu\in F}a_\mu\ch T_p(\mu), \qquad a_\mu\in\CC, \end{equation} where every \(\mu\in F\) satisfies \[ V_{u_q}(Q(T_p(\mu)))\subseteq\cC. \] Discard the zero coefficients. Every weight in a highest-weight module of highest weight \(\nu\) is at most \(\nu\), and its highest-weight multiplicity is one. If a weight \(\mu\) in \(F\) were not at most \(\lambda\), choose one maximal for the dominance order. The coefficient of \(e^\mu\) in the right-hand side of \eqref{end:triangular} would be \(a_\mu\ne0\), whereas it is zero in the left-hand side. This is impossible. Thus every \(\mu\in F\) is at most \(\lambda\). Comparison of the coefficient of \(e^\lambda\) then gives \(a_\lambda=1\). Consequently \begin{equation}\label{end:upper} V_{u_q}(Q(T_p(\lambda))) \subseteq\overline{\cO_\lambda}. \end{equation} The quantum tilting module \(T_q(\lambda)\) occurs with multiplicity one in \(Q(T_p(\lambda))\): this follows again from the unique highest weight and its multiplicity. The quantum support theorem gives the reverse inclusion in \eqref{end:upper}. Finally Theorem~\ref{lift:whole} yields \[ V_{G_1}(T_p(\lambda))=\overline{\cO_\lambda}. \] \subsection{Singular weights} Let \(\lambda\in X^+\) be arbitrary, and let \(w\) label its lower-closure alcove as in the introduction. The translation principle for tilting tensor ideals gives \[ V_{G_1}(T_p(\lambda)) =V_{G_1}(T_p(w\bullet_p0)); \] see \cite[Lemma~7.6]{AHR}. We recall the mechanism relevant to this equality. Translation to the face containing \(\lambda\) and translation out of that face are linkage summands of tensoring with tilting modules. The lower-closure convention selects the tilting summand in the regular block, and the two tiltings belong to each other's tensor ideals. Taking a summand can only decrease support, and tensoring intersects supports. The two inclusions therefore give equality. The regular weight \(w\bullet_p0\) has the orbit label \(\cO_w=\cO_\lambda\), so the established regular formula proves the theorem for \(\lambda\) as well. \subsection{Products, central quotients, and the connected center} For a product of simply connected simple groups, indecomposable tilting modules are exterior tensor products. The cohomological K\"unneth isomorphism identifies their supports with products of the supports in the factors. Ordinary affine cells and their nilpotent orbits have the same product decomposition. The theorem therefore holds for every semisimple simply connected group. Now let \(G\) be connected reductive, let \(G_{\mathrm{der}}^{\mathrm{sc}}\) be the simply connected cover of its derived group, and let \(Z^\circ\) be its connected center. The structure theorem for reductive groups \cite[Corollary~5.3.3 and Proposition~5.3.4]{Conrad} gives a central isogeny \begin{equation}\label{end:isogeny} f:G_{\mathrm{der}}^{\mathrm{sc}}\times Z^\circ\longrightarrow G. \end{equation} Its kernel injects into the center of \(G_{\mathrm{der}}^{\mathrm{sc}}\). The primes dividing the order of that center are at most the Coxeter number of one of its simple factors. Since \(p>h\), the kernel of \(f\) is finite etale. Frobenius is an isomorphism on this kernel, and the induced map on first Frobenius kernels is an isomorphism \[ (G_{\mathrm{der}}^{\mathrm{sc}})_1\times(Z^\circ)_1 \xrightarrow{\sim}G_1. \] The differential of \(f\) identifies the nilpotent cones in the derived Lie algebras and preserves their orbit labels. Pullback under the surjective map \(f\) is fully faithful. It takes Weyl and dual Weyl modules to the corresponding modules for the covering root datum, so it takes an indecomposable tilting module to the indecomposable tilting module with the pulled-back highest weight. Such a module is a tilting module for \(G_{\mathrm{der}}^{\mathrm{sc}}\) tensored with a character of \(Z^\circ\). The infinitesimal torus \((Z^\circ)_1\) is diagonalizable and linearly reductive; its positive-degree cohomology vanishes. Thus this torus factor contributes no nonzero direction to the support. Applying the simply connected formula and the isomorphism of Frobenius kernels proves Theorem~\ref{main} for \(G\). If \(G\) itself is a torus, both its nilpotent cone and every nonzero module support are the single point \(0\), giving the same conclusion. \begin{corollary}\label{end:transition} If the quantum tilting module \(T_q(\mu)\) occurs as a summand of the whole quantum lift \(Q(T_p(\lambda))\), then \[ \cO_\mu\subseteq\overline{\cO_\lambda}. \] \end{corollary} \begin{proof} The support of a direct summand is contained in the support of the whole module. Apply the quantum support theorem, Theorem~\ref{lift:whole}, and Theorem~\ref{main}. \end{proof} % End sections/conclusion.tex \clearpage \appendix % Begin sections/certificates.tex \section{Explicit finite certificates}\label{app:certificates} This appendix specifies the finite computations used in Section~\ref{ex:section}. The certificate consists of reduced words, integer Laurent polynomials, integral representation matrices, exact character calculations in cyclotomic fields, and finite orbit down-sets. The rules below determine the mathematical meaning and the verification of every record. The accompanying electronic files give the larger integer arrays in machine-readable form and implement these rules. \subsection{Roots, words, and spherical canonical vectors} \label{app:canonical} For $E_r$, let $A=(a_{ij})$ be the Cartan matrix with diagonal $2$ and off-diagonal entries $-1$ on the edges listed at the start of Section~\ref{ex:section}. On the column basis $e_1,\ldots,e_r$ set \begin{equation}\label{app:root-matrices} S_i=I-e_i(a_{i1},\ldots,a_{ir}). \end{equation} Starting with $\{e_1,\ldots,e_r\}$ and closing under the $S_i$ gives the roots. The positive roots are those with nonnegative coordinates; there are $63$ and $120$ for $r=7$ and $8$. For a word $a=i_1\cdots i_d$ write $S(a)=S_{i_1}\cdots S_{i_d}$. Its element has length \[ \ell(S(a))=\#\{\alpha>0:S(a)\alpha<0\}. \] Thus reducedness is an integer matrix test. Right $W_J$-minimality is the test $S(a)e_j>0$ for every $j\in J$. Repeated removal of right descents in $J$ gives the unique minimal coset representative and the accompanying parabolic factor. Repeated addition of right ascents in $J$ constructs $w_J$. Put $m_x=H_xC_{w_J}$ for $x\in W^J$. If $s$ is simple, let $x'$ be the minimal element of $sxW_J$. The elementary spherical rule is \begin{equation}\label{app:spherical-multiplication} C_sm_x= \begin{cases} m_{x'}+vm_x,&\ell(x')=\ell(x)+1,\\ m_{x'}+v^{-1}m_x,&\ell(x')=\ell(x)-1,\\ (v+v^{-1})m_x,&x'=x. \end{cases} \end{equation} This rule and the integer root matrices suffice for all the spherical polynomial calculations in the paper. For each $x$, the vector $\mathbf c_x=C_{xw_J}$ is uniquely specified by \begin{equation}\label{app:canonical-tests} \overline{\mathbf c_x}=\mathbf c_x,\qquad \mathbf c_x=m_x+\sum_{y2d$, the integer trace of a finite-order matrix is the unique integer in $[-d,d]$ with its computed residue. This makes the class-trace check exact. To construct $e_J$, choose a chain of standard parabolics \[ 1=W_{J_0}\subset W_{J_1}\subset\cdots\subset W_{J_t}=W_J \] and a set $D_a$ of minimal coset representatives at each step. The products of the $D_a$ give each element of $W_J$ once, so \[ e_J=\prod_{a=t}^{1}\left(\sum_{d\in D_a}R(d)\right) \] with the order chosen to match that coset decomposition. The certificate lists the factors as words. Their distinctness, minimality, and cardinalities are verified by the root matrices. Starting with the seed vector $v_0$ defined in Section~\ref{ex:matrices}, this computes the integer vector $u=e_Jv_0$. One checks $R_j u=u$ for all $j\in J$. Finally, specialize the independently defined spherical vector in (\ref{app:canonical-tests}) at $v=1$ and evaluate (\ref{ex:action-formula}). The resulting exact coordinates give Table~\ref{ex:e8-actions}. The largest conservative intermediate integer bounds in these three vector calculations are, respectively, $133497216$, $123604992$, and $34384896$. Alternatively arbitrary precision integer arithmetic gives the same calculation without an a priori bound. A nonzero coordinate already proves the required block detection; the displayed sum of squares supplies an additional exact check on the entire vector. For the two restriction calculations, use (\ref{ex:highest-reflection}) to construct $R_0$. The sign projectors are averages over groups of orders $6$ and $12$. Their images have dimensions $10$ and $5$, respectively. On a basis of each image, form the linear equations \[ XR_i-R_iX=0 \] for the remaining simple generators. The coefficient matrices have nullities one modulo $1000003$, hence also over $\QQ$. The character identifications and determinant conversion are given in Lemma~\ref{ex:last-two}. This verifies both multiplicities directly from the integral matrices. \subsection{The finite trace-and-cut verification rules} \label{app:finite} We now specify the objects occurring in the source and witness tables. The affine node $0$ is represented in the finite Weyl group by reflection in the highest root. If $b$ is a mask in rank $r$, write \[ b=\sum_{j=1}^{r}\epsilon_j2^{j-1}+\epsilon_0 2^r, \qquad \epsilon_j\in\{0,1\}. \] Then $H_b$ is generated by $s_j$ with $\epsilon_j=1$ and by $s_\theta$ when $\epsilon_0=1$. Only proper affine node subsets are used. Thus the mask specifies the embedding of the subgroup. For example mask $255$ is the finite $E_8$, and mask $510$ is its affine $D_8$ subgroup. For a mask not using node $0$, take $\xi=0$. Otherwise let $k$ be the least missing finite node and set \begin{equation}\label{app:fixed-weight} \langle\xi,\alpha_i^\vee\rangle=\delta_{ik}/m_k, \end{equation} where $\theta^\vee=\sum_i m_i\alpha_i^\vee$. This is a fixed point for the corresponding finite affine parabolic. Its lift is exactly $t_{\xi-h\xi}h$. One checks $\xi-h\xi$ is integral for every generator and hence every $h$. This fixes all phases in (\ref{ex:phase-trace}). Here are explicit conventions for the vertex representatives. Let $\omega_i^\vee$ be the fundamental coweights and put $y_i=\exp(2\pi\mathrm i\,\omega_i^\vee/m_i)$ for $i>0$, $y_0=1$. The highest-root coefficient lists are \[ (m_i)_{E_7}=(2,2,3,4,3,2,1),\qquad (m_i)_{E_8}=(2,3,4,6,5,4,3,2). \] In simple-coroot coordinates, the representative is the $i$th column of $A^{-1}$ divided by $m_i$, taken modulo $\ZZ^r$. The tables list the vertices in the following order; the entries are the deleted-node indices: \begin{equation}\label{app:vertex-order} \begin{split} E_7:&\quad (0,7,6,1,3,4,2,5),\\ E_8:&\quad (0,1,8,7,2,6,3,5,4). \end{split} \end{equation} Deleting that node from the extended diagram gives its centralizer. The representatives retain all central translates in the simply connected datum. The full root systems of the centralizers can also be recovered without diagram recognition: take exactly the roots $\alpha$ for which $\alpha(y)=1$ and their reflections. In the product-orbit notation of the witness tables, the ordered factors are as follows: \[ \begin{array}{c|l} E_7\text{ vertices}& E_7, E_7, D_6A_1, D_6A_1, A_2A_5, A_3A_1A_3, A_7, A_5A_2\\ E_8\text{ vertices}& E_8, D_8, E_7A_1, E_6A_2, A_8, D_5A_3, A_1A_7, A_4A_4, A_2A_1A_5 . \end{array} \] Components are ordered by their least finite node; the affine node is placed last for this purpose. Within a component use its standard based root system. Positive and negative very-even labels distinguish the two orbits in the usual type-$D$ Springer tables. A string of single digits denotes a partition; a part larger than nine is written in a parenthesized list. Commas between factors denote products. The label $1$ in an exceptional orbit column denotes the zero orbit. The finite input for a subgroup is its ordinary character table, class representative words and centralizer orders, the Springer pairs, the $a$- and $b$-values, and the orbit closure incidence relation. These are the standard finite Weyl-group and Springer data in \cite{LusztigIV,GeckPfeiffer,GH}. Character row labels in the source tables are standard $\phi_{d,b}$ labels in exceptional type, partitions in type $A$, and bipartitions separated by a dot in type $D$; signs distinguish a degenerate pair. Product characters are written as lists of their factor labels. The additional row number fixes the ordering of the accompanying character array, but the character label and embedded subgroup specify the representation independently of that number. For each row of a witness table perform the following finite steps. \begin{enumerate}[label=\arabic*.,leftmargin=2em] \item Construct its vertex point $y$, its centralizer $M$, and the irreducible $E$ corresponding to its printed local nilpotent orbit with trivial local system. Embed its weighted Dynkin diagram in the ambient root space, make it dominant, and check that it gives the printed ambient orbit. \item Read the indicated source ID in Table~\ref{app:blocks-7} or~\ref{app:blocks-8}. Construct $H_b$ and $\xi$ by (\ref{app:fixed-weight}). For each conjugacy-class representative $h$ of $H_b$, enumerate the points $z\in Wy$ fixed by $h$, and evaluate (\ref{ex:phase-trace}). Combine these values with the ordinary character of $F$ using class sizes to obtain (\ref{ex:trace-multiplicity}). Reduce cyclotomic polynomials modulo the $e$-th cyclotomic polynomial $\Phi_e$, with $e$ the order of $y$. The resulting value is a strictly positive integer. Check nonnegativity and integrality for the full multiplicity column and the degree identity (\ref{ex:degree-check}). \item For every ordinary parabolic $L$ of $M$, compute $[\Res_L^M E:\det_L]$ by character inner product. Keep exactly the positive cases. Compute $j_L^W\det_L$: among the constituents of $\Ind_L^W\det_L$, it is the unique one with $b=b(\det_L)$, and its multiplicity is one. Read its Springer orbit $D_L$. \item In the finite orbit incidence relation intersect the down-set of the source bound with the down-sets of all retained $D_L$. Every element of this intersection lies in the down-set of the printed target. Check also that the target itself lies in the source down-set. These are precisely the two assertions in (\ref{ex:finite-test}). \end{enumerate} Parabolic subgroups in the third step may be taken up to conjugacy in $M$, because both the restriction multiplicity and the ambient orbit are constant on such a class. The identity mask and the trivial subgroup are included. The group-intersection step in the trace calculation uses all fixed points and all conjugated elements $r_zhr_z^{-1}$ in the full intersection. For source ID $8.\mathrm G$, Step~2 can instead use the explicit multiplicity-one calculation of Lemma~\ref{ex:last-two}; Step~4 then holds because its source bound is already the target. The electronic implementation separates three kinds of input. The integral word and representation records are verified by matrix multiplication and Laurent-polynomial identities. The ordinary character and Springer arrays are finite classification data; their row and column identifications, degrees, class orthogonality, induction dimensions, and closure down-sets fix the conventions. The cyclotomic trace and cut routines evaluate the four steps just given from these inputs. The published exhaustive $p$-canonical calculations supply the precise rank bound stated in Section~\ref{ex:section}. The propagation and geometric arguments establish actuality of the remaining sources. \subsection{Selected actual finite blocks} \label{app:block-tables} Tables~\ref{app:blocks-7} and~\ref{app:blocks-8} list the distinct ordinary blocks used by the coverage calculation. A source ID has the form $r.k$ in type $E_r$ and identifies a row of this table. For each ordinary block $F$, its bound is obtained from the special representation of its family by truncated induction from the embedded $H_b$ to $W(E_r)$. The exceptional actuality certificates are Proposition~\ref{ex:actual} and Theorem~\ref{e8:source}; the uniform blocks are supplied by the finite, minuscule, and classical constructions described in Section~\ref{ex:section}. % Begin sections/exceptional_blocks_7.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{lr r p{3.7cm} l} \caption{Selected actual finite blocks in $E_7$. The mask defines $H$; $F$ is in the ordinary Springer convention.}\label{app:blocks-7}\\ \toprule ID & Mask & Row & $F$ & Bound $B_F$\\\midrule \endfirsthead \toprule ID & Mask & Row & $F$ & Bound $B_F$\\\midrule \endhead \bottomrule\endfoot 7.001 & 0 & 1 & \texttt{Id} & $E_7$\\ 7.002 & 13 & 1 & \texttt{1111} & $D_6(a_1)$\\ 7.003 & 15 & 1 & \texttt{11111} & $D_5(a_1)$\\ 7.004 & 19 & 1 & \texttt{11,11,11} & $E_7(a_3)$\\ 7.005 & 23 & 1 & \texttt{111,11,11} & $E_7(a_4)$\\ 7.006 & 27 & 1 & \texttt{11,1111} & $E_7(a_5)$\\ 7.007 & 30 & 3 & \texttt{1.111} & $E_7(a_5)$\\ 7.008 & 30 & 4 & \texttt{.1111} & $A_5''$\\ 7.009 & 31 & 2 & \texttt{1.1111} & $A_4$\\ 7.010 & 31 & 3 & \texttt{.11111} & $(A_3+A_1)''$\\ 7.011 & 47 & 1 & \texttt{11111,11} & $A_4+A_1$\\ 7.012 & 53 & 1 & \texttt{111,111} & $D_5+A_1$\\ 7.013 & 55 & 1 & \texttt{111,11,111} & $E_7(a_5)$\\ 7.014 & 61 & 1 & \texttt{111111} & $D_4(a_1)+A_1$\\ 7.015 & 63 & 2 & \texttt{phi1,36} & $3A_1''$\\ 7.016 & 63 & 5 & \texttt{phi6,25} & $A_2+A_1$\\ 7.017 & 63 & 8 & \texttt{phi15,17} & $D_4(a_1)+A_1$\\ 7.018 & 83 & 1 & \texttt{11,11,11,11} & $E_6(a_1)$\\ 7.019 & 87 & 1 & \texttt{111,11,11,11} & $A_6$\\ 7.020 & 90 & 1 & \texttt{1111,11} & $D_5$\\ 7.021 & 91 & 1 & \texttt{11,1111,11} & $E_6(a_3)$\\ 7.022 & 94 & 7 & \texttt{.1111,11} & $A_4$\\ 7.023 & 95 & 3 & \texttt{1.1111,11} & $A_3+A_2$\\ 7.024 & 95 & 5 & \texttt{.11111,11} & $2A_2$\\ 7.025 & 109 & 1 & \texttt{1111,111} & $D_5(a_1)+A_1$\\ 7.026 & 111 & 1 & \texttt{11111,111} & $A_3+A_2+A_1$\\ 7.027 & 119 & 1 & \texttt{111,11,1111} & $A_4+A_2$\\ 7.028 & 122 & 1 & \texttt{111111} & $D_4$\\ 7.029 & 123 & 1 & \texttt{11,111111} & $D_4(a_1)$\\ 7.030 & 125 & 1 & \texttt{1111111} & $A_2+3A_1$\\ 7.031 & 126 & 3 & \texttt{11.1111} & $D_4(a_1)$\\ 7.032 & 126 & 4 & \texttt{1.11111} & $A_3$\\ 7.033 & 126 & 5 & \texttt{.111111} & $A_2$\\ 7.034 & 126 & 18 & \texttt{.222} & $D_5(a_1)$\\ 7.035 & 127 & 2 & \texttt{phi1,63} & $1$\\ 7.036 & 127 & 3 & \texttt{phi7,46} & $A_1$\\ 7.037 & 127 & 5 & \texttt{phi15,28} & $A_2+A_1$\\ 7.038 & 127 & 12 & \texttt{phi27,37} & $2A_1$\\ 7.039 & 127 & 35 & \texttt{phi189,22} & $A_2+2A_1$\\ 7.040 & 212 & 1 & \texttt{11,11,11,11} & $D_6$\\ 7.041 & 213 & 1 & \texttt{1111,11,11} & $D_6(a_2)$\\ 7.042 & 215 & 1 & \texttt{1111,11,11,11} & $A_5'$\\ 7.043 & 221 & 1 & \texttt{111111,11} & $A_3+2A_1$\\ 7.044 & 222 & 9 & \texttt{1.111,11,11} & $D_5(a_1)+A_1$\\ 7.045 & 222 & 13 & \texttt{.1111,11,11} & $A_3+A_2$\\ 7.046 & 223 & 5 & \texttt{11.1111,11} & $(A_3+A_1)'$\\ 7.047 & 223 & 7 & \texttt{1.11111,11} & $A_2+2A_1$\\ 7.048 & 223 & 9 & \texttt{.111111,11} & $3A_1'$\\ 7.049 & 223 & 35 & \texttt{.222,11} & $A_4+A_1$\\ 7.050 & 235 & 1 & \texttt{111,111,111} & $A_5+A_1$\\ 7.051 & 239 & 1 & \texttt{111111,111} & $2A_2+A_1$\\ 7.052 & 245 & 1 & \texttt{1111,1111} & $D_4+A_1$\\ 7.053 & 247 & 1 & \texttt{1111,11,1111} & $A_3+A_2+A_1$\\ 7.054 & 253 & 1 & \texttt{11111111} & $4A_1$\\ \end{longtable} \endgroup % End sections/exceptional_blocks_7.tex % Begin sections/exceptional_blocks_8.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{lr r p{3.7cm} l} \caption{Selected actual finite blocks in $E_8$. The mask defines $H$; $F$ is in the ordinary Springer convention.}\label{app:blocks-8}\\ \toprule ID & Mask & Row & $F$ & Bound $B_F$\\\midrule \endfirsthead \toprule ID & Mask & Row & $F$ & Bound $B_F$\\\midrule \endhead \bottomrule\endfoot 8.001 & 0 & 1 & \texttt{Id} & $E_8$\\ 8.002 & 13 & 1 & \texttt{1111} & $E_7(a_1)$\\ 8.003 & 15 & 1 & \texttt{11111} & $E_7(a_3)$\\ 8.004 & 19 & 1 & \texttt{11,11,11} & $E_8(a_3)$\\ 8.005 & 23 & 1 & \texttt{111,11,11} & $E_8(b_4)$\\ 8.006 & 27 & 1 & \texttt{11,1111} & $E_8(b_5)$\\ 8.007 & 30 & 3 & \texttt{1.111} & $E_8(b_5)$\\ 8.008 & 30 & 4 & \texttt{.1111} & $E_6$\\ 8.009 & 31 & 2 & \texttt{1.1111} & $E_6(a_1)$\\ 8.010 & 31 & 3 & \texttt{.11111} & $D_5$\\ 8.011 & 47 & 1 & \texttt{11111,11} & $E_6(a_1)+A_1$\\ 8.012 & 55 & 1 & \texttt{111,11,111} & $E_8(b_5)$\\ 8.013 & 61 & 1 & \texttt{111111} & $D_6(a_1)$\\ 8.014 & 63 & 2 & \texttt{phi1,36} & $D_4$\\ 8.015 & 63 & 5 & \texttt{phi6,25} & $D_5(a_1)$\\ 8.016 & 63 & 8 & \texttt{phi15,17} & $D_6(a_1)$\\ 8.017 & 83 & 1 & \texttt{11,11,11,11} & $E_8(a_4)$\\ 8.018 & 87 & 1 & \texttt{111,11,11,11} & $E_8(a_5)$\\ 8.019 & 91 & 1 & \texttt{11,1111,11} & $E_8(a_6)$\\ 8.020 & 94 & 5 & \texttt{1.111,11} & $E_8(a_6)$\\ 8.021 & 94 & 7 & \texttt{.1111,11} & $E_6(a_1)$\\ 8.022 & 95 & 3 & \texttt{1.1111,11} & $E_7(a_4)$\\ 8.023 & 95 & 5 & \texttt{.11111,11} & $E_6(a_3)$\\ 8.024 & 109 & 1 & \texttt{1111,111} & $D_7(a_1)$\\ 8.025 & 111 & 1 & \texttt{11111,111} & $D_5+A_2$\\ 8.026 & 119 & 1 & \texttt{111,11,1111} & $E_8(b_6)$\\ 8.027 & 123 & 1 & \texttt{11,111111} & $E_8(a_7)$\\ 8.028 & 125 & 1 & \texttt{1111111} & $D_4+A_2$\\ 8.029 & 126 & 3 & \texttt{11.1111} & $E_8(a_7)$\\ 8.030 & 126 & 4 & \texttt{1.11111} & $E_6(a_3)$\\ 8.031 & 126 & 5 & \texttt{.111111} & $A_4$\\ 8.032 & 126 & 18 & \texttt{.222} & $E_7(a_3)$\\ 8.033 & 127 & 2 & \texttt{phi1,63} & $A_2$\\ 8.034 & 127 & 3 & \texttt{phi7,46} & $A_3$\\ 8.035 & 127 & 5 & \texttt{phi15,28} & $D_5(a_1)$\\ 8.036 & 127 & 12 & \texttt{phi27,37} & $D_4(a_1)$\\ 8.037 & 127 & 35 & \texttt{phi189,22} & $D_5(a_1)+A_1$\\ 8.038 & 175 & 1 & \texttt{11111,11,11} & $D_7(a_2)$\\ 8.039 & 183 & 1 & \texttt{111,11,111,11} & $E_8(a_6)$\\ 8.040 & 191 & 3 & \texttt{phi1,36,11} & $D_4(a_1)$\\ 8.041 & 191 & 9 & \texttt{phi6,25,11} & $A_4+A_1$\\ 8.042 & 191 & 15 & \texttt{phi15,17,11} & $E_8(a_7)$\\ 8.043 & 222 & 7 & \texttt{1.111,111} & $E_8(b_6)$\\ 8.044 & 222 & 10 & \texttt{.1111,111} & $A_6$\\ 8.045 & 223 & 4 & \texttt{1.1111,111} & $E_8(a_7)$\\ 8.046 & 223 & 7 & \texttt{.11111,111} & $A_4+A_2$\\ 8.047 & 237 & 1 & \texttt{1111,1111} & $D_7(a_2)$\\ 8.048 & 247 & 1 & \texttt{111,11,11111} & $A_6+A_1$\\ 8.049 & 251 & 1 & \texttt{11,1111111} & $A_4+A_2+A_1$\\ 8.050 & 253 & 1 & \texttt{11111111} & $D_4(a_1)+A_2$\\ 8.051 & 254 & 2 & \texttt{11.11111} & $A_4+2A_1$\\ 8.052 & 254 & 3 & \texttt{1.111111} & $A_3+A_2$\\ 8.053 & 254 & 4 & \texttt{.1111111} & $2A_2$\\ 8.054 & 255 & 2 & \texttt{phi1,120} & $1$\\ 8.055 & 255 & 6 & \texttt{phi35,74} & $2A_1$\\ 8.056 & 255 & 9 & \texttt{phi50,56} & $A_2+A_1$\\ 8.057 & 255 & 16 & \texttt{phi210,52} & $A_2+A_1$\\ 8.058 & 255 & 38 & \texttt{phi1050,34} & $D_4(a_1)+A_1$\\ 8.059 & 255 & 44 & \texttt{phi1400,32} & $D_4(a_1)+A_1$\\ 8.060 & 255 & 69 & \texttt{phi8,91} & $A_1$\\ 8.061 & 255 & 82 & \texttt{phi560,47} & $A_2+2A_1$\\ 8.062 & 338 & 1 & \texttt{11,11,11,11} & $E_7$\\ 8.063 & 343 & 1 & \texttt{111,11,11,11,11} & $D_7$\\ 8.064 & 346 & 1 & \texttt{1111,11,11} & $E_7(a_2)$\\ 8.065 & 347 & 1 & \texttt{11,1111,11,11} & $D_7(a_1)$\\ 8.066 & 350 & 9 & \texttt{1.111,11,11} & $D_7(a_1)$\\ 8.067 & 350 & 13 & \texttt{.1111,11,11} & $E_7(a_4)$\\ 8.068 & 351 & 5 & \texttt{1.1111,11,11} & $A_6$\\ 8.069 & 351 & 9 & \texttt{.11111,11,11} & $A_5$\\ 8.070 & 375 & 1 & \texttt{111,11,1111,11} & $A_7$\\ 8.071 & 378 & 1 & \texttt{111111,11} & $D_5+A_1$\\ 8.072 & 379 & 1 & \texttt{11,111111,11} & $E_7(a_5)$\\ 8.073 & 382 & 5 & \texttt{11.1111,11} & $E_7(a_5)$\\ 8.074 & 382 & 7 & \texttt{1.11111,11} & $D_5(a_1)+A_1$\\ 8.075 & 382 & 9 & \texttt{.111111,11} & $A_3+A_2$\\ 8.076 & 382 & 10 & \texttt{.111111,2} & $A_4$\\ 8.077 & 382 & 35 & \texttt{.222,11} & $E_6(a_1)+A_1$\\ 8.078 & 383 & 3 & \texttt{phi1,63,11} & $3A_1$\\ 8.079 & 383 & 4 & \texttt{phi1,63,2} & $A_2$\\ 8.080 & 383 & 5 & \texttt{phi7,46,11} & $A_2+2A_1$\\ 8.081 & 383 & 6 & \texttt{phi7,46,2} & $A_3$\\ 8.082 & 383 & 9 & \texttt{phi15,28,11} & $A_4+A_1$\\ 8.083 & 383 & 17 & \texttt{phi21,36,11} & $D_4(a_1)$\\ 8.084 & 383 & 23 & \texttt{phi27,37,11} & $A_3+A_1$\\ 8.085 & 383 & 69 & \texttt{phi189,22,11} & $A_4+A_2$\\ 8.086 & 437 & 1 & \texttt{111,111,111} & $E_6+A_1$\\ 8.087 & 445 & 1 & \texttt{111111,111} & $E_6(a_3)+A_1$\\ 8.088 & 447 & 4 & \texttt{phi1,36,111} & $2A_2+A_1$\\ 8.089 & 447 & 13 & \texttt{phi6,25,111} & $D_4(a_1)+A_2$\\ 8.090 & 447 & 14 & \texttt{phi6,25,21} & $A_4+A_1$\\ 8.091 & 447 & 22 & \texttt{phi15,17,111} & $E_6(a_3)+A_1$\\ 8.092 & 447 & 28 & \texttt{phi15,16,111} & $E_6(a_3)+A_1$\\ 8.093 & 447 & 29 & \texttt{phi15,16,21} & $E_8(a_7)$\\ 8.094 & 447 & 35 & \texttt{phi20,20,21} & $E_6(a_3)$\\ 8.095 & 474 & 1 & \texttt{1111,1111} & $D_6$\\ 8.096 & 475 & 1 & \texttt{11,1111,1111} & $D_5+A_2$\\ 8.097 & 478 & 16 & \texttt{.1111,1111} & $D_6(a_2)$\\ 8.098 & 479 & 1 & \texttt{11.111,1111} & $E_8(a_7)$\\ 8.099 & 479 & 6 & \texttt{1.1111,1111} & $D_5(a_1)+A_2$\\ 8.100 & 479 & 11 & \texttt{.11111,1111} & $2A_3$\\ 8.101 & 479 & 13 & \texttt{.11111,22} & $A_5$\\ 8.102 & 495 & 1 & \texttt{11111,11111} & $A_4+A_3$\\ 8.103 & 503 & 1 & \texttt{111,11,111111} & $A_5+A_1$\\ 8.104 & 503 & 23 & \texttt{21,11,111111} & $E_7(a_5)$\\ 8.105 & 506 & 1 & \texttt{11111111} & $D_4+A_1$\\ 8.106 & 507 & 1 & \texttt{11,11111111} & $A_3+A_2+A_1$\\ 8.107 & 507 & 2 & \texttt{11,2111111} & $A_4+A_2+A_1$\\ 8.108 & 509 & 1 & \texttt{111111111} & $2A_2+2A_1$\\ 8.109 & 509 & 2 & \texttt{21111111} & $D_4(a_1)+A_2$\\ 8.110 & 510 & 3 & \texttt{111.11111} & $A_3+A_2+A_1$\\ 8.111 & 510 & 4 & \texttt{11.111111} & $A_3+2A_1$\\ 8.112 & 510 & 5 & \texttt{1.1111111} & $A_2+3A_1$\\ 8.113 & 510 & 6 & \texttt{.11111111} & $4A_1$\\ 8.114 & 510 & 28 & \texttt{.22211} & $A_4+2A_1$\\ 8.G & 255 & 14 & $\phi_{175,36}$ & $D_4(a_1)+A_1$\\ \end{longtable} \endgroup % End sections/exceptional_blocks_8.tex \subsection{All required local witnesses in type \texorpdfstring{$E_7$}{E7}} \label{app:e7-witnesses} For a local orbit in the following tables, the local system is always trivial. The Springer correspondence therefore determines the local irreducible. The row number identifies the same entry in the full ordinary character array; gaps in those row numbers are the omitted nontrivial-local-system entries. Each displayed source satisfies the positive trace and cut-intersection conditions of (\ref{ex:finite-test}). % Begin sections/exceptional_witnesses_7.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 1, $M=E_7$: all 45 trivial-local-system generators.}\label{app:witness-7-1}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $E_7$ & $E_7$ & 7.001\\ 2 & $1$ & $1$ & 7.001\\ 3 & $A_1$ & $A_1$ & 7.036\\ 4 & $E_7(a_1)$ & $E_7(a_1)$ & 7.001\\ 5 & $4A_1$ & $4A_1$ & 7.054\\ 9 & $3A_1''$ & $3A_1''$ & 7.001\\ 10 & $E_6$ & $E_6$ & 7.001\\ 11 & $E_7(a_2)$ & $E_7(a_2)$ & 7.001\\ 12 & $2A_1$ & $2A_1$ & 7.001\\ 15 & $D_6$ & $D_6$ & 7.040\\ 16 & $3A_1'$ & $3A_1'$ & 7.048\\ 17 & $A_2$ & $A_2$ & 7.001\\ 18 & $E_7(a_3)$ & $E_7(a_3)$ & 7.001\\ 19 & $2A_2+A_1$ & $2A_2+A_1$ & 7.051\\ 20 & $A_5+A_1$ & $A_5+A_1$ & 7.050\\ 21 & $D_4+A_1$ & $D_4+A_1$ & 7.052\\ 25 & $A_6$ & $A_6$ & 7.001\\ 26 & $A_2+3A_1$ & $A_2+3A_1$ & 7.001\\ 27 & $A_5''$ & $A_5''$ & 7.001\\ 28 & $D_4$ & $D_4$ & 7.001\\ 29 & $E_6(a_1)$ & $E_6(a_1)$ & 7.001\\ 30 & $A_2+A_1$ & $A_2+A_1$ & 7.016\\ 31 & $D_5+A_1$ & $D_5+A_1$ & 7.001\\ 32 & $2A_2$ & $2A_2$ & 7.001\\ 35 & $A_2+2A_1$ & $A_2+2A_1$ & 7.001\\ 36 & $E_7(a_4)$ & $E_7(a_4)$ & 7.001\\ 37 & $(A_3+A_1)''$ & $(A_3+A_1)''$ & 7.001\\ 38 & $D_5$ & $D_5$ & 7.001\\ 39 & $D_6(a_1)$ & $D_6(a_1)$ & 7.001\\ 40 & $A_3$ & $A_3$ & 7.032\\ 41 & $A_4+A_2$ & $A_4+A_2$ & 7.001\\ 42 & $A_3+A_2+A_1$ & $A_3+A_2+A_1$ & 7.001\\ 43 & $A_3+2A_1$ & $A_3+2A_1$ & 7.043\\ 44 & $A_5'$ & $A_5'$ & 7.042\\ 47 & $D_6(a_2)$ & $D_6(a_2)$ & 7.041\\ 48 & $(A_3+A_1)'$ & $(A_3+A_1)'$ & 7.043\\ 49 & $D_4(a_1)$ & $D_4(a_1)$ & 7.001\\ 50 & $E_7(a_5)$ & $E_7(a_5)$ & 7.001\\ 53 & $A_3+A_2$ & $A_3+A_2$ & 7.001\\ 54 & $D_5(a_1)+A_1$ & $D_5(a_1)+A_1$ & 7.001\\ 55 & $E_6(a_3)$ & $E_6(a_3)$ & 7.001\\ 56 & $D_4(a_1)+A_1$ & $D_4(a_1)+A_1$ & 7.001\\ 57 & $D_5(a_1)$ & $D_5(a_1)$ & 7.001\\ 58 & $A_4$ & $A_4$ & 7.001\\ 60 & $A_4+A_1$ & $A_4+A_1$ & 7.001\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 2, $M=E_7$: all 45 trivial-local-system generators.}\label{app:witness-7-2}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $E_7$ & $E_7$ & 7.001\\ 2 & $1$ & $1$ & 7.001\\ 3 & $A_1$ & $A_1$ & 7.036\\ 4 & $E_7(a_1)$ & $E_7(a_1)$ & 7.001\\ 5 & $4A_1$ & $4A_1$ & 7.054\\ 9 & $3A_1''$ & $3A_1''$ & 7.001\\ 10 & $E_6$ & $E_6$ & 7.001\\ 11 & $E_7(a_2)$ & $E_7(a_2)$ & 7.001\\ 12 & $2A_1$ & $2A_1$ & 7.001\\ 15 & $D_6$ & $D_6$ & 7.040\\ 16 & $3A_1'$ & $3A_1'$ & 7.048\\ 17 & $A_2$ & $A_2$ & 7.001\\ 18 & $E_7(a_3)$ & $E_7(a_3)$ & 7.001\\ 19 & $2A_2+A_1$ & $2A_2+A_1$ & 7.051\\ 20 & $A_5+A_1$ & $A_5+A_1$ & 7.050\\ 21 & $D_4+A_1$ & $D_4+A_1$ & 7.052\\ 25 & $A_6$ & $A_6$ & 7.001\\ 26 & $A_2+3A_1$ & $A_2+3A_1$ & 7.001\\ 27 & $A_5''$ & $A_5''$ & 7.001\\ 28 & $D_4$ & $D_4$ & 7.001\\ 29 & $E_6(a_1)$ & $E_6(a_1)$ & 7.001\\ 30 & $A_2+A_1$ & $A_2+A_1$ & 7.016\\ 31 & $D_5+A_1$ & $D_5+A_1$ & 7.001\\ 32 & $2A_2$ & $2A_2$ & 7.001\\ 35 & $A_2+2A_1$ & $A_2+2A_1$ & 7.001\\ 36 & $E_7(a_4)$ & $E_7(a_4)$ & 7.001\\ 37 & $(A_3+A_1)''$ & $(A_3+A_1)''$ & 7.001\\ 38 & $D_5$ & $D_5$ & 7.001\\ 39 & $D_6(a_1)$ & $D_6(a_1)$ & 7.001\\ 40 & $A_3$ & $A_3$ & 7.032\\ 41 & $A_4+A_2$ & $A_4+A_2$ & 7.001\\ 42 & $A_3+A_2+A_1$ & $A_3+A_2+A_1$ & 7.001\\ 43 & $A_3+2A_1$ & $A_3+2A_1$ & 7.043\\ 44 & $A_5'$ & $A_5'$ & 7.042\\ 47 & $D_6(a_2)$ & $D_6(a_2)$ & 7.041\\ 48 & $(A_3+A_1)'$ & $(A_3+A_1)'$ & 7.043\\ 49 & $D_4(a_1)$ & $D_4(a_1)$ & 7.001\\ 50 & $E_7(a_5)$ & $E_7(a_5)$ & 7.001\\ 53 & $A_3+A_2$ & $A_3+A_2$ & 7.001\\ 54 & $D_5(a_1)+A_1$ & $D_5(a_1)+A_1$ & 7.001\\ 55 & $E_6(a_3)$ & $E_6(a_3)$ & 7.001\\ 56 & $D_4(a_1)+A_1$ & $D_4(a_1)+A_1$ & 7.001\\ 57 & $D_5(a_1)$ & $D_5(a_1)$ & 7.001\\ 58 & $A_4$ & $A_4$ & 7.001\\ 60 & $A_4+A_1$ & $A_4+A_1$ & 7.001\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 3, $M=D_6+A_1$: all 62 trivial-local-system generators.}\label{app:witness-7-3}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $222222+,11$ & $3A_1'$ & 7.048\\ 2 & $222222+,2$ & $A_2$ & 7.033\\ 3 & $222222-,11$ & $3A_1''$ & 7.015\\ 4 & $222222-,2$ & $4A_1$ & 7.054\\ 5 & $22221111,11$ & $2A_1$ & 7.038\\ 6 & $22221111,2$ & $3A_1'$ & 7.048\\ 7 & $2211111111,11$ & $A_1$ & 7.036\\ 8 & $2211111111,2$ & $2A_1$ & 7.038\\ 9 & $111111111111,11$ & $1$ & 7.035\\ 10 & $111111111111,2$ & $A_1$ & 7.036\\ 15 & $332211,11$ & $A_2+A_1$ & 7.016\\ 16 & $332211,2$ & $A_2+2A_1$ & 7.039\\ 17 & $33111111,11$ & $A_2$ & 7.033\\ 18 & $33111111,2$ & $A_2+A_1$ & 7.016\\ 19 & $3111111111,11$ & $2A_1$ & 7.038\\ 20 & $3111111111,2$ & $3A_1''$ & 7.015\\ 21 & $3333,11$ & $2A_2$ & 7.024\\ 22 & $3333,2$ & $2A_2+A_1$ & 7.051\\ 23 & $4422+,11$ & $(A_3+A_1)'$ & 7.046\\ 24 & $4422+,2$ & $D_4(a_1)$ & 7.029\\ 25 & $4422-,11$ & $(A_3+A_1)''$ & 7.010\\ 26 & $4422-,2$ & $A_3+2A_1$ & 7.043\\ 27 & $333111,11$ & $A_2+2A_1$ & 7.039\\ 28 & $333111,2$ & $A_2+3A_1$ & 7.030\\ 29 & $441111,11$ & $A_3$ & 7.032\\ 30 & $441111,2$ & $(A_3+A_1)'$ & 7.046\\ 31 & $32211111,11$ & $3A_1'$ & 7.048\\ 32 & $32211111,2$ & $4A_1$ & 7.054\\ 33 & $4431,11$ & $A_3+2A_1$ & 7.043\\ 34 & $4431,2$ & $D_4(a_1)+A_1$ & 7.014\\ 35 & $322221,11$ & $4A_1$ & 7.054\\ 36 & $322221,2$ & $A_2+A_1$ & 7.037\\ 39 & $5322,11$ & $D_4(a_1)+A_1$ & 7.014\\ 40 & $5322,2$ & $A_3+A_2$ & 7.023\\ 41 & $531111,11$ & $D_4(a_1)$ & 7.029\\ 42 & $531111,2$ & $D_4(a_1)+A_1$ & 7.014\\ 43 & $51111111,11$ & $A_3$ & 7.032\\ 44 & $51111111,2$ & $(A_3+A_1)''$ & 7.010\\ 47 & $5331,11$ & $A_3+A_2$ & 7.023\\ 48 & $5331,2$ & $A_3+A_2+A_1$ & 7.026\\ 49 & $5511,11$ & $A_4$ & 7.009\\ 50 & $5511,2$ & $A_4+A_1$ & 7.011\\ 51 & $522111,11$ & $(A_3+A_1)'$ & 7.046\\ 52 & $522111,2$ & $A_3+2A_1$ & 7.043\\ 53 & $66+,11$ & $A_5'$ & 7.042\\ 54 & $66+,2$ & $E_6(a_3)$ & 7.021\\ 55 & $66-,11$ & $A_5''$ & 7.008\\ 56 & $66-,2$ & $A_5+A_1$ & 7.050\\ 61 & $7311,11$ & $D_5(a_1)$ & 7.003\\ 62 & $7311,2$ & $D_5(a_1)+A_1$ & 7.025\\ 63 & $711111,11$ & $D_4$ & 7.028\\ 64 & $711111,2$ & $D_4+A_1$ & 7.052\\ 65 & $75,11$ & $D_6(a_2)$ & 7.041\\ 66 & $75,2$ & $E_7(a_5)$ & 7.006\\ 67 & $7221,11$ & $D_4+A_1$ & 7.052\\ 68 & $7221,2$ & $D_5(a_1)$ & 7.034\\ 69 & $93,11$ & $D_6(a_1)$ & 7.002\\ 70 & $93,2$ & $E_7(a_4)$ & 7.005\\ 71 & $9111,11$ & $D_5$ & 7.020\\ 72 & $9111,2$ & $D_5+A_1$ & 7.012\\ 73 & $(11,1),11$ & $D_6$ & 7.040\\ 74 & $(11,1),2$ & $E_7(a_3)$ & 7.004\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 4, $M=D_6+A_1$: all 62 trivial-local-system generators.}\label{app:witness-7-4}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $222222+,11$ & $3A_1''$ & 7.015\\ 2 & $222222+,2$ & $4A_1$ & 7.054\\ 3 & $222222-,11$ & $3A_1'$ & 7.048\\ 4 & $222222-,2$ & $A_2$ & 7.033\\ 5 & $22221111,11$ & $2A_1$ & 7.038\\ 6 & $22221111,2$ & $3A_1'$ & 7.048\\ 7 & $2211111111,11$ & $A_1$ & 7.036\\ 8 & $2211111111,2$ & $2A_1$ & 7.038\\ 9 & $111111111111,11$ & $1$ & 7.035\\ 10 & $111111111111,2$ & $A_1$ & 7.036\\ 15 & $332211,11$ & $A_2+A_1$ & 7.016\\ 16 & $332211,2$ & $A_2+2A_1$ & 7.039\\ 17 & $33111111,11$ & $A_2$ & 7.033\\ 18 & $33111111,2$ & $A_2+A_1$ & 7.016\\ 19 & $3111111111,11$ & $2A_1$ & 7.038\\ 20 & $3111111111,2$ & $3A_1''$ & 7.015\\ 21 & $3333,11$ & $2A_2$ & 7.024\\ 22 & $3333,2$ & $2A_2+A_1$ & 7.051\\ 23 & $4422+,11$ & $(A_3+A_1)''$ & 7.010\\ 24 & $4422+,2$ & $A_3+2A_1$ & 7.043\\ 25 & $4422-,11$ & $(A_3+A_1)'$ & 7.046\\ 26 & $4422-,2$ & $D_4(a_1)$ & 7.029\\ 27 & $333111,11$ & $A_2+2A_1$ & 7.039\\ 28 & $333111,2$ & $A_2+3A_1$ & 7.030\\ 29 & $441111,11$ & $A_3$ & 7.032\\ 30 & $441111,2$ & $(A_3+A_1)'$ & 7.046\\ 31 & $32211111,11$ & $3A_1'$ & 7.048\\ 32 & $32211111,2$ & $4A_1$ & 7.054\\ 33 & $4431,11$ & $A_3+2A_1$ & 7.043\\ 34 & $4431,2$ & $D_4(a_1)+A_1$ & 7.014\\ 35 & $322221,11$ & $4A_1$ & 7.054\\ 36 & $322221,2$ & $A_2+A_1$ & 7.037\\ 39 & $5322,11$ & $D_4(a_1)+A_1$ & 7.014\\ 40 & $5322,2$ & $A_3+A_2$ & 7.023\\ 41 & $531111,11$ & $D_4(a_1)$ & 7.029\\ 42 & $531111,2$ & $D_4(a_1)+A_1$ & 7.014\\ 43 & $51111111,11$ & $A_3$ & 7.032\\ 44 & $51111111,2$ & $(A_3+A_1)''$ & 7.010\\ 47 & $5331,11$ & $A_3+A_2$ & 7.023\\ 48 & $5331,2$ & $A_3+A_2+A_1$ & 7.026\\ 49 & $5511,11$ & $A_4$ & 7.009\\ 50 & $5511,2$ & $A_4+A_1$ & 7.011\\ 51 & $522111,11$ & $(A_3+A_1)'$ & 7.046\\ 52 & $522111,2$ & $A_3+2A_1$ & 7.043\\ 53 & $66+,11$ & $A_5''$ & 7.008\\ 54 & $66+,2$ & $A_5+A_1$ & 7.050\\ 55 & $66-,11$ & $A_5'$ & 7.042\\ 56 & $66-,2$ & $E_6(a_3)$ & 7.021\\ 61 & $7311,11$ & $D_5(a_1)$ & 7.003\\ 62 & $7311,2$ & $D_5(a_1)+A_1$ & 7.025\\ 63 & $711111,11$ & $D_4$ & 7.028\\ 64 & $711111,2$ & $D_4+A_1$ & 7.052\\ 65 & $75,11$ & $D_6(a_2)$ & 7.041\\ 66 & $75,2$ & $E_7(a_5)$ & 7.006\\ 67 & $7221,11$ & $D_4+A_1$ & 7.052\\ 68 & $7221,2$ & $D_5(a_1)$ & 7.034\\ 69 & $93,11$ & $D_6(a_1)$ & 7.002\\ 70 & $93,2$ & $E_7(a_4)$ & 7.005\\ 71 & $9111,11$ & $D_5$ & 7.020\\ 72 & $9111,2$ & $D_5+A_1$ & 7.012\\ 73 & $(11,1),11$ & $D_6$ & 7.040\\ 74 & $(11,1),2$ & $E_7(a_3)$ & 7.004\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 5, $M=A_5+A_2$: all 33 trivial-local-system generators.}\label{app:witness-7-5}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $111,111111$ & $1$ & 7.035\\ 2 & $111,21111$ & $A_1$ & 7.036\\ 3 & $111,2211$ & $2A_1$ & 7.038\\ 4 & $111,222$ & $3A_1''$ & 7.015\\ 5 & $111,3111$ & $A_2$ & 7.033\\ 6 & $111,321$ & $A_2+A_1$ & 7.016\\ 7 & $111,33$ & $2A_2$ & 7.024\\ 8 & $111,411$ & $A_3$ & 7.032\\ 9 & $111,42$ & $(A_3+A_1)''$ & 7.010\\ 10 & $111,51$ & $A_4$ & 7.009\\ 11 & $111,6$ & $A_5''$ & 7.008\\ 12 & $21,111111$ & $A_1$ & 7.036\\ 13 & $21,21111$ & $2A_1$ & 7.038\\ 14 & $21,2211$ & $3A_1'$ & 7.048\\ 15 & $21,222$ & $4A_1$ & 7.054\\ 16 & $21,3111$ & $A_2+A_1$ & 7.016\\ 17 & $21,321$ & $A_2+2A_1$ & 7.039\\ 18 & $21,33$ & $2A_2+A_1$ & 7.051\\ 19 & $21,411$ & $(A_3+A_1)'$ & 7.046\\ 20 & $21,42$ & $A_3+2A_1$ & 7.043\\ 21 & $21,51$ & $A_4+A_1$ & 7.011\\ 22 & $21,6$ & $A_5+A_1$ & 7.050\\ 23 & $3,111111$ & $A_2$ & 7.033\\ 24 & $3,21111$ & $A_2+A_1$ & 7.016\\ 25 & $3,2211$ & $A_2+2A_1$ & 7.039\\ 26 & $3,222$ & $A_2+3A_1$ & 7.030\\ 27 & $3,3111$ & $2A_2$ & 7.024\\ 28 & $3,321$ & $2A_2+A_1$ & 7.051\\ 29 & $3,33$ & $D_4(a_1)$ & 7.029\\ 30 & $3,411$ & $A_3+A_2$ & 7.023\\ 31 & $3,42$ & $A_3+A_2+A_1$ & 7.026\\ 32 & $3,51$ & $A_4+A_2$ & 7.027\\ 33 & $3,6$ & $E_7(a_5)$ & 7.013\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 6, $M=2A_3+A_1$: all 50 trivial-local-system generators.}\label{app:witness-7-6}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $1111,11,1111$ & $1$ & 7.035\\ 2 & $1111,11,211$ & $A_1$ & 7.036\\ 3 & $1111,11,22$ & $2A_1$ & 7.038\\ 4 & $1111,11,31$ & $A_2$ & 7.033\\ 5 & $1111,11,4$ & $A_3$ & 7.032\\ 6 & $1111,2,1111$ & $A_1$ & 7.036\\ 7 & $1111,2,211$ & $2A_1$ & 7.038\\ 8 & $1111,2,22$ & $3A_1''$ & 7.015\\ 9 & $1111,2,31$ & $A_2+A_1$ & 7.016\\ 10 & $1111,2,4$ & $(A_3+A_1)''$ & 7.010\\ 11 & $211,11,1111$ & $A_1$ & 7.036\\ 12 & $211,11,211$ & $2A_1$ & 7.038\\ 13 & $211,11,22$ & $3A_1'$ & 7.048\\ 14 & $211,11,31$ & $A_2+A_1$ & 7.016\\ 15 & $211,11,4$ & $(A_3+A_1)'$ & 7.046\\ 16 & $211,2,1111$ & $2A_1$ & 7.038\\ 17 & $211,2,211$ & $3A_1'$ & 7.048\\ 18 & $211,2,22$ & $4A_1$ & 7.054\\ 19 & $211,2,31$ & $A_2+2A_1$ & 7.039\\ 20 & $211,2,4$ & $A_3+2A_1$ & 7.043\\ 21 & $22,11,1111$ & $2A_1$ & 7.038\\ 22 & $22,11,211$ & $3A_1'$ & 7.048\\ 23 & $22,11,22$ & $A_2$ & 7.033\\ 24 & $22,11,31$ & $A_2+2A_1$ & 7.039\\ 25 & $22,11,4$ & $D_4(a_1)$ & 7.029\\ 26 & $22,2,1111$ & $3A_1''$ & 7.015\\ 27 & $22,2,211$ & $4A_1$ & 7.054\\ 28 & $22,2,22$ & $A_2+A_1$ & 7.037\\ 29 & $22,2,31$ & $A_2+3A_1$ & 7.030\\ 30 & $22,2,4$ & $D_4(a_1)+A_1$ & 7.014\\ 31 & $31,11,1111$ & $A_2$ & 7.033\\ 32 & $31,11,211$ & $A_2+A_1$ & 7.016\\ 33 & $31,11,22$ & $A_2+2A_1$ & 7.039\\ 34 & $31,11,31$ & $2A_2$ & 7.024\\ 35 & $31,11,4$ & $A_3+A_2$ & 7.023\\ 36 & $31,2,1111$ & $A_2+A_1$ & 7.016\\ 37 & $31,2,211$ & $A_2+2A_1$ & 7.039\\ 38 & $31,2,22$ & $A_2+3A_1$ & 7.030\\ 39 & $31,2,31$ & $2A_2+A_1$ & 7.051\\ 40 & $31,2,4$ & $A_3+A_2+A_1$ & 7.026\\ 41 & $4,11,1111$ & $A_3$ & 7.032\\ 42 & $4,11,211$ & $(A_3+A_1)'$ & 7.046\\ 43 & $4,11,22$ & $D_4(a_1)$ & 7.029\\ 44 & $4,11,31$ & $A_3+A_2$ & 7.023\\ 45 & $4,11,4$ & $A_4$ & 7.022\\ 46 & $4,2,1111$ & $(A_3+A_1)''$ & 7.010\\ 47 & $4,2,211$ & $A_3+2A_1$ & 7.043\\ 48 & $4,2,22$ & $D_4(a_1)+A_1$ & 7.014\\ 49 & $4,2,31$ & $A_3+A_2+A_1$ & 7.026\\ 50 & $4,2,4$ & $A_4+A_1$ & 7.049\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 7, $M=A_7$: all 22 trivial-local-system generators.}\label{app:witness-7-7}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $11111111$ & $1$ & 7.035\\ 2 & $2111111$ & $A_1$ & 7.036\\ 3 & $221111$ & $2A_1$ & 7.038\\ 4 & $22211$ & $3A_1'$ & 7.048\\ 5 & $2222$ & $A_2$ & 7.033\\ 6 & $311111$ & $A_2$ & 7.033\\ 7 & $32111$ & $A_2+A_1$ & 7.016\\ 8 & $3221$ & $A_2+2A_1$ & 7.039\\ 9 & $3311$ & $2A_2$ & 7.024\\ 10 & $332$ & $2A_2+A_1$ & 7.051\\ 11 & $41111$ & $A_3$ & 7.032\\ 12 & $4211$ & $(A_3+A_1)'$ & 7.046\\ 13 & $422$ & $D_4(a_1)$ & 7.029\\ 14 & $431$ & $A_3+A_2$ & 7.023\\ 15 & $44$ & $A_4$ & 7.022\\ 16 & $5111$ & $A_4$ & 7.009\\ 17 & $521$ & $A_4+A_1$ & 7.011\\ 18 & $53$ & $A_4+A_2$ & 7.027\\ 19 & $611$ & $A_5'$ & 7.042\\ 20 & $62$ & $E_6(a_3)$ & 7.021\\ 21 & $71$ & $A_6$ & 7.019\\ 22 & $8$ & $E_6(a_1)$ & 7.018\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_7$, vertex 8, $M=A_5+A_2$: all 33 trivial-local-system generators.}\label{app:witness-7-8}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $111111,111$ & $1$ & 7.035\\ 2 & $111111,21$ & $A_1$ & 7.036\\ 3 & $111111,3$ & $A_2$ & 7.033\\ 4 & $21111,111$ & $A_1$ & 7.036\\ 5 & $21111,21$ & $2A_1$ & 7.038\\ 6 & $21111,3$ & $A_2+A_1$ & 7.016\\ 7 & $2211,111$ & $2A_1$ & 7.038\\ 8 & $2211,21$ & $3A_1'$ & 7.048\\ 9 & $2211,3$ & $A_2+2A_1$ & 7.039\\ 10 & $222,111$ & $3A_1''$ & 7.015\\ 11 & $222,21$ & $4A_1$ & 7.054\\ 12 & $222,3$ & $A_2+3A_1$ & 7.030\\ 13 & $3111,111$ & $A_2$ & 7.033\\ 14 & $3111,21$ & $A_2+A_1$ & 7.016\\ 15 & $3111,3$ & $2A_2$ & 7.024\\ 16 & $321,111$ & $A_2+A_1$ & 7.016\\ 17 & $321,21$ & $A_2+2A_1$ & 7.039\\ 18 & $321,3$ & $2A_2+A_1$ & 7.051\\ 19 & $33,111$ & $2A_2$ & 7.024\\ 20 & $33,21$ & $2A_2+A_1$ & 7.051\\ 21 & $33,3$ & $D_4(a_1)$ & 7.029\\ 22 & $411,111$ & $A_3$ & 7.032\\ 23 & $411,21$ & $(A_3+A_1)'$ & 7.046\\ 24 & $411,3$ & $A_3+A_2$ & 7.023\\ 25 & $42,111$ & $(A_3+A_1)''$ & 7.010\\ 26 & $42,21$ & $A_3+2A_1$ & 7.043\\ 27 & $42,3$ & $A_3+A_2+A_1$ & 7.026\\ 28 & $51,111$ & $A_4$ & 7.009\\ 29 & $51,21$ & $A_4+A_1$ & 7.011\\ 30 & $51,3$ & $A_4+A_2$ & 7.027\\ 31 & $6,111$ & $A_5''$ & 7.008\\ 32 & $6,21$ & $A_5+A_1$ & 7.050\\ 33 & $6,3$ & $E_7(a_5)$ & 7.013\\ \end{longtable} \endgroup % End sections/exceptional_witnesses_7.tex \subsection{All required local witnesses in type \texorpdfstring{$E_8$}{E8}} \label{app:e8-witnesses} The conventions are the same as for $E_7$. In particular the two entries with source $8.\mathrm G$ are the two explicit multiplicity-one restrictions in Lemma~\ref{ex:last-two}. % Begin sections/exceptional_witnesses_8.tex % Generated by certificates/generate_exceptional_tables.py. \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 1, $M=E_8$: all 70 trivial-local-system generators.}\label{app:witness-8-1}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $E_8$ & $E_8$ & 8.001\\ 2 & $1$ & $1$ & 8.001\\ 5 & $E_8(a_2)$ & $E_8(a_2)$ & 8.001\\ 6 & $2A_1$ & $2A_1$ & 8.055\\ 9 & $4A_1$ & $4A_1$ & 8.113\\ 10 & $E_7$ & $E_7$ & 8.062\\ 11 & $3A_1$ & $3A_1$ & 8.078\\ 14 & $2A_2+2A_1$ & $2A_2+2A_1$ & 8.108\\ 15 & $E_8(a_4)$ & $E_8(a_4)$ & 8.001\\ 16 & $A_2+A_1$ & $A_2+A_1$ & 8.057\\ 17 & $A_4+A_3$ & $A_4+A_3$ & 8.102\\ 22 & $E_6$ & $E_6$ & 8.001\\ 23 & $D_4$ & $D_4$ & 8.001\\ 24 & $E_7(a_1)$ & $E_7(a_1)$ & 8.001\\ 25 & $A_3$ & $A_3$ & 8.034\\ 28 & $D_4+A_1$ & $D_4+A_1$ & 8.105\\ 29 & $E_8(a_5)$ & $E_8(a_5)$ & 8.001\\ 30 & $2A_2$ & $2A_2$ & 8.001\\ 33 & $2A_3$ & $2A_3$ & 8.100\\ 35 & $D_6$ & $D_6$ & 8.095\\ 38 & $A_3+2A_1$ & $A_3+2A_1$ & 8.111\\ 39 & $D_5$ & $D_5$ & 8.001\\ 40 & $E_7(a_2)$ & $E_7(a_2)$ & 8.064\\ 41 & $A_3+A_1$ & $A_3+A_1$ & 8.084\\ 43 & $E_8(a_6)$ & $E_8(a_6)$ & 8.001\\ 44 & $D_4(a_1)+A_1$ & $D_4(a_1)+A_1$ & 8.059\\ 47 & $E_6(a_3)+A_1$ & $E_6(a_3)+A_1$ & 8.087\\ 50 & $D_6(a_2)$ & $D_6(a_2)$ & 8.097\\ 51 & $E_8(b_6)$ & $E_8(b_6)$ & 8.001\\ 52 & $D_4(a_1)+A_2$ & $D_4(a_1)+A_2$ & 8.001\\ 53 & $E_8(a_7)$ & $E_8(a_7)$ & 8.001\\ 54 & $E_7(a_3)$ & $E_7(a_3)$ & 8.001\\ 55 & $A_4$ & $A_4$ & 8.001\\ 57 & $A_6+A_1$ & $A_6+A_1$ & 8.001\\ 58 & $A_4+A_2+A_1$ & $A_4+A_2+A_1$ & 8.001\\ 60 & $D_5+A_1$ & $D_5+A_1$ & 8.071\\ 61 & $A_5$ & $A_5$ & 8.069\\ 63 & $A_4+A_1$ & $A_4+A_1$ & 8.015\\ 64 & $D_7(a_2)$ & $D_7(a_2)$ & 8.001\\ 65 & $A_4+2A_1$ & $A_4+2A_1$ & 8.051\\ 66 & $E_7(a_4)$ & $E_7(a_4)$ & 8.001\\ 67 & $D_5(a_1)+A_1$ & $D_5(a_1)+A_1$ & 8.001\\ 68 & $E_8(a_1)$ & $E_8(a_1)$ & 8.001\\ 69 & $A_1$ & $A_1$ & 8.060\\ 72 & $E_8(a_3)$ & $E_8(a_3)$ & 8.001\\ 73 & $A_2$ & $A_2$ & 8.001\\ 77 & $D_7$ & $D_7$ & 8.063\\ 78 & $A_2+3A_1$ & $A_2+3A_1$ & 8.112\\ 79 & $E_6+A_1$ & $E_6+A_1$ & 8.086\\ 80 & $2A_2+A_1$ & $2A_2+A_1$ & 8.088\\ 81 & $E_8(b_4)$ & $E_8(b_4)$ & 8.001\\ 82 & $A_2+2A_1$ & $A_2+2A_1$ & 8.061\\ 83 & $D_5(a_1)+A_2$ & $D_5(a_1)+A_2$ & 8.099\\ 88 & $A_5+A_1$ & $A_5+A_1$ & 8.103\\ 91 & $A_7$ & $A_7$ & 8.070\\ 92 & $A_3+A_2+A_1$ & $A_3+A_2+A_1$ & 8.105\\ 93 & $E_8(b_5)$ & $E_8(b_5)$ & 8.001\\ 94 & $D_4(a_1)$ & $D_4(a_1)$ & 8.001\\ 97 & $E_6(a_1)$ & $E_6(a_1)$ & 8.001\\ 98 & $D_5(a_1)$ & $D_5(a_1)$ & 8.015\\ 100 & $D_7(a_1)$ & $D_7(a_1)$ & 8.001\\ 101 & $A_3+A_2$ & $A_3+A_2$ & 8.001\\ 104 & $E_7(a_5)$ & $E_7(a_5)$ & 8.071\\ 105 & $E_6(a_1)+A_1$ & $E_6(a_1)+A_1$ & 8.001\\ 107 & $A_6$ & $A_6$ & 8.001\\ 108 & $D_4+A_2$ & $D_4+A_2$ & 8.001\\ 109 & $D_5+A_2$ & $D_5+A_2$ & 8.001\\ 110 & $A_4+A_2$ & $A_4+A_2$ & 8.001\\ 111 & $D_6(a_1)$ & $D_6(a_1)$ & 8.001\\ 112 & $E_6(a_3)$ & $E_6(a_3)$ & 8.001\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 2, $M=D_8$: all 75 trivial-local-system generators.}\label{app:witness-8-2}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $22222222-$ & $4A_1$ & 8.113\\ 2 & $22222222+$ & $A_2$ & 8.033\\ 3 & $2222221111$ & $3A_1$ & 8.078\\ 4 & $222211111111$ & $2A_1$ & 8.055\\ 5 & $22111111111111$ & $A_1$ & 8.060\\ 6 & $1111111111111111$ & $1$ & 8.054\\ 10 & $33222211$ & $A_2+2A_1$ & 8.061\\ 11 & $3322111111$ & $A_2+A_1$ & 8.057\\ 12 & $331111111111$ & $A_2$ & 8.033\\ 13 & $31111111111111$ & $2A_1$ & 8.055\\ 15 & $333322$ & $2A_2+A_1$ & 8.088\\ 16 & $442222-$ & $A_3+2A_1$ & 8.111\\ 17 & $442222+$ & $D_4(a_1)$ & 8.036\\ 18 & $33331111$ & $2A_2$ & 8.053\\ 19 & $44221111$ & $A_3+A_1$ & 8.084\\ 20 & $3331111111$ & $A_2+2A_1$ & 8.061\\ 21 & $4411111111$ & $A_3$ & 8.034\\ 22 & $322111111111$ & $3A_1$ & 8.078\\ 24 & $333331$ & $2A_2+2A_1$ & 8.108\\ 25 & $443311$ & $A_3+A_2$ & 8.052\\ 26 & $33322111$ & $A_2+3A_1$ & 8.112\\ 27 & $44311111$ & $A_3+2A_1$ & 8.111\\ 28 & $3222211111$ & $4A_1$ & 8.113\\ 29 & $4444-$ & $2A_3$ & 8.100\\ 30 & $4444+$ & $A_4$ & 8.031\\ 31 & $443221$ & $D_4(a_1)+A_1$ & 8.058\\ 32 & $32222221$ & $A_2+A_1$ & 8.056\\ 35 & $532222$ & $A_3+A_2$ & 8.052\\ 36 & $53221111$ & $D_4(a_1)+A_1$ & 8.059\\ 37 & $5311111111$ & $D_4(a_1)$ & 8.036\\ 38 & $511111111111$ & $A_3$ & 8.034\\ 42 & $533311$ & $D_4(a_1)+A_2$ & 8.050\\ 43 & $552211$ & $A_4+A_1$ & 8.041\\ 44 & $53311111$ & $A_3+A_2$ & 8.052\\ 45 & $55111111$ & $A_4$ & 8.031\\ 46 & $5221111111$ & $A_3+A_1$ & 8.084\\ 48 & $5533$ & $A_4+A_2$ & 8.046\\ 49 & $533221$ & $A_3+A_2+A_1$ & 8.110\\ 51 & $553111$ & $A_4+2A_1$ & 8.051\\ 52 & $52222111$ & $A_3+2A_1$ & 8.111\\ 53 & $5443$ & $A_4+2A_1$ & 8.114\\ 54 & $6622-$ & $A_5+A_1$ & 8.103\\ 55 & $6622+$ & $E_6(a_3)$ & 8.023\\ 56 & $544111$ & $2A_3$ & 8.100\\ 57 & $661111$ & $A_5$ & 8.069\\ 59 & $5551$ & $A_4+A_3$ & 8.102\\ 60 & $6631$ & $E_6(a_3)+A_1$ & 8.087\\ 64 & $732211$ & $D_5(a_1)+A_1$ & 8.037\\ 65 & $73111111$ & $D_5(a_1)$ & 8.015\\ 66 & $7111111111$ & $D_4$ & 8.014\\ 68 & $7333$ & $D_5(a_1)+A_2$ & 8.099\\ 69 & $7522$ & $E_7(a_5)$ & 8.072\\ 70 & $733111$ & $D_4+A_2$ & 8.028\\ 71 & $751111$ & $D_6(a_2)$ & 8.097\\ 72 & $72211111$ & $D_4+A_1$ & 8.105\\ 74 & $7531$ & $E_8(a_7)$ & 8.027\\ 75 & $722221$ & $D_5(a_1)$ & 8.035\\ 77 & $7441$ & $D_6(a_2)$ & 8.097\\ 78 & $7711$ & $A_6$ & 8.044\\ 80 & $88-$ & $A_7$ & 8.070\\ 81 & $88+$ & $E_6(a_1)$ & 8.021\\ 84 & $9322$ & $E_7(a_4)$ & 8.022\\ 85 & $931111$ & $D_6(a_1)$ & 8.013\\ 86 & $91111111$ & $D_5$ & 8.010\\ 88 & $9331$ & $D_5+A_2$ & 8.025\\ 89 & $9511$ & $D_7(a_2)$ & 8.038\\ 90 & $922111$ & $D_5+A_1$ & 8.071\\ 91 & $97$ & $E_8(b_6)$ & 8.026\\ 94 & $(11,3,1,1)$ & $D_7(a_1)$ & 8.024\\ 95 & $(11,1,1,1,1,1)$ & $D_6$ & 8.095\\ 96 & $(11,5)$ & $E_8(a_6)$ & 8.019\\ 97 & $(11,2,2,1)$ & $E_7(a_3)$ & 8.032\\ 98 & $(13,3)$ & $E_8(a_5)$ & 8.018\\ 99 & $(13,1,1,1)$ & $D_7$ & 8.063\\ 100 & $(15,1)$ & $E_8(a_4)$ & 8.017\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 3, $M=E_7+A_1$: all 90 trivial-local-system generators.}\label{app:witness-8-3}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $E_7,11$ & $E_7$ & 8.062\\ 2 & $E_7,2$ & $E_8(a_3)$ & 8.004\\ 3 & $1,11$ & $1$ & 8.054\\ 4 & $1,2$ & $A_1$ & 8.060\\ 5 & $A_1,11$ & $A_1$ & 8.060\\ 6 & $A_1,2$ & $2A_1$ & 8.055\\ 7 & $E_7(a_1),11$ & $E_7(a_1)$ & 8.002\\ 8 & $E_7(a_1),2$ & $E_8(b_4)$ & 8.005\\ 9 & $4A_1,11$ & $4A_1$ & 8.113\\ 10 & $4A_1,2$ & $A_2+A_1$ & 8.056\\ 17 & $3A_1'',11$ & $3A_1$ & 8.078\\ 18 & $3A_1'',2$ & $A_2$ & 8.033\\ 19 & $E_6,11$ & $E_6$ & 8.008\\ 20 & $E_6,2$ & $E_6+A_1$ & 8.086\\ 21 & $E_7(a_2),11$ & $E_7(a_2)$ & 8.064\\ 22 & $E_7(a_2),2$ & $E_8(b_5)$ & 8.006\\ 23 & $2A_1,11$ & $2A_1$ & 8.055\\ 24 & $2A_1,2$ & $3A_1$ & 8.078\\ 29 & $D_6,11$ & $D_6$ & 8.095\\ 30 & $D_6,2$ & $E_7(a_3)$ & 8.032\\ 31 & $3A_1',11$ & $3A_1$ & 8.078\\ 32 & $3A_1',2$ & $4A_1$ & 8.113\\ 33 & $A_2,11$ & $A_2$ & 8.033\\ 34 & $A_2,2$ & $A_2+A_1$ & 8.057\\ 35 & $E_7(a_3),11$ & $E_7(a_3)$ & 8.003\\ 36 & $E_7(a_3),2$ & $D_7(a_1)$ & 8.024\\ 37 & $2A_2+A_1,11$ & $2A_2+A_1$ & 8.088\\ 38 & $2A_2+A_1,2$ & $2A_2+2A_1$ & 8.108\\ 39 & $A_5+A_1,11$ & $A_5+A_1$ & 8.103\\ 40 & $A_5+A_1,2$ & $E_6(a_3)+A_1$ & 8.087\\ 41 & $D_4+A_1,11$ & $D_4+A_1$ & 8.105\\ 42 & $D_4+A_1,2$ & $D_5(a_1)$ & 8.035\\ 49 & $A_6,11$ & $A_6$ & 8.044\\ 50 & $A_6,2$ & $A_6+A_1$ & 8.048\\ 51 & $A_2+3A_1,11$ & $A_2+3A_1$ & 8.112\\ 52 & $A_2+3A_1,2$ & $2A_2$ & 8.053\\ 53 & $A_5'',11$ & $A_5$ & 8.069\\ 54 & $A_5'',2$ & $E_6(a_3)$ & 8.023\\ 55 & $D_4,11$ & $D_4$ & 8.014\\ 56 & $D_4,2$ & $D_4+A_1$ & 8.105\\ 57 & $E_6(a_1),11$ & $E_6(a_1)$ & 8.009\\ 58 & $E_6(a_1),2$ & $E_6(a_1)+A_1$ & 8.011\\ 59 & $A_2+A_1,11$ & $A_2+A_1$ & 8.057\\ 60 & $A_2+A_1,2$ & $A_2+2A_1$ & 8.061\\ 61 & $D_5+A_1,11$ & $D_5+A_1$ & 8.071\\ 62 & $D_5+A_1,2$ & $D_6(a_1)$ & 8.013\\ 63 & $2A_2,11$ & $2A_2$ & 8.053\\ 64 & $2A_2,2$ & $2A_2+A_1$ & 8.088\\ 69 & $A_2+2A_1,11$ & $A_2+2A_1$ & 8.061\\ 70 & $A_2+2A_1,2$ & $A_2+3A_1$ & 8.112\\ 71 & $E_7(a_4),11$ & $E_7(a_4)$ & 8.022\\ 72 & $E_7(a_4),2$ & $D_5+A_2$ & 8.025\\ 73 & $(A_3+A_1)'',11$ & $A_3+A_1$ & 8.084\\ 74 & $(A_3+A_1)'',2$ & $D_4(a_1)$ & 8.036\\ 75 & $D_5,11$ & $D_5$ & 8.010\\ 76 & $D_5,2$ & $D_5+A_1$ & 8.071\\ 77 & $D_6(a_1),11$ & $D_6(a_1)$ & 8.013\\ 78 & $D_6(a_1),2$ & $E_7(a_4)$ & 8.022\\ 79 & $A_3,11$ & $A_3$ & 8.034\\ 80 & $A_3,2$ & $A_3+A_1$ & 8.084\\ 81 & $A_4+A_2,11$ & $A_4+A_2$ & 8.046\\ 82 & $A_4+A_2,2$ & $A_4+A_2+A_1$ & 8.049\\ 83 & $A_3+A_2+A_1,11$ & $A_3+A_2+A_1$ & 8.106\\ 84 & $A_3+A_2+A_1,2$ & $D_4(a_1)+A_2$ & 8.050\\ 85 & $A_3+2A_1,11$ & $A_3+2A_1$ & 8.111\\ 86 & $A_3+2A_1,2$ & $D_4(a_1)+A_1$ & 8.058\\ 87 & $A_5',11$ & $A_5$ & 8.069\\ 88 & $A_5',2$ & $A_5+A_1$ & 8.103\\ 93 & $D_6(a_2),11$ & $D_6(a_2)$ & 8.097\\ 94 & $D_6(a_2),2$ & $E_7(a_5)$ & 8.072\\ 95 & $(A_3+A_1)',11$ & $A_3+A_1$ & 8.084\\ 96 & $(A_3+A_1)',2$ & $A_3+2A_1$ & 8.111\\ 97 & $D_4(a_1),11$ & $D_4(a_1)$ & 8.036\\ 98 & $D_4(a_1),2$ & $D_4(a_1)+A_1$ & 8.059\\ 99 & $E_7(a_5),11$ & $E_7(a_5)$ & 8.072\\ 100 & $E_7(a_5),2$ & $E_8(a_7)$ & 8.027\\ 105 & $A_3+A_2,11$ & $A_3+A_2$ & 8.052\\ 106 & $A_3+A_2,2$ & $A_3+A_2+A_1$ & 8.110\\ 107 & $D_5(a_1)+A_1,11$ & $D_5(a_1)+A_1$ & 8.037\\ 108 & $D_5(a_1)+A_1,2$ & $D_4+A_2$ & 8.028\\ 109 & $E_6(a_3),11$ & $E_6(a_3)$ & 8.023\\ 110 & $E_6(a_3),2$ & $E_6(a_3)+A_1$ & 8.087\\ 111 & $D_4(a_1)+A_1,11$ & $D_4(a_1)+A_1$ & 8.059\\ 112 & $D_4(a_1)+A_1,2$ & $A_3+A_2$ & 8.052\\ 113 & $D_5(a_1),11$ & $D_5(a_1)$ & 8.015\\ 114 & $D_5(a_1),2$ & $D_5(a_1)+A_1$ & 8.037\\ 115 & $A_4,11$ & $A_4$ & 8.031\\ 116 & $A_4,2$ & $A_4+A_1$ & 8.041\\ 119 & $A_4+A_1,11$ & $A_4+A_1$ & 8.041\\ 120 & $A_4+A_1,2$ & $A_4+2A_1$ & 8.051\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 4, $M=E_6+A_2$: all 63 trivial-local-system generators.}\label{app:witness-8-4}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $E_6,111$ & $E_6$ & 8.008\\ 2 & $E_6,21$ & $E_6+A_1$ & 8.086\\ 3 & $E_6,3$ & $E_8(b_5)$ & 8.012\\ 4 & $1,111$ & $1$ & 8.054\\ 5 & $1,21$ & $A_1$ & 8.060\\ 6 & $1,3$ & $A_2$ & 8.033\\ 7 & $2A_2+A_1,111$ & $2A_2+A_1$ & 8.088\\ 8 & $2A_2+A_1,21$ & $2A_2+2A_1$ & 8.108\\ 9 & $2A_2+A_1,3$ & $D_4(a_1)+A_1$ & 8.G\\ 10 & $E_6(a_1),111$ & $E_6(a_1)$ & 8.009\\ 11 & $E_6(a_1),21$ & $E_6(a_1)+A_1$ & 8.011\\ 12 & $E_6(a_1),3$ & $E_8(b_6)$ & 8.026\\ 13 & $A_1,111$ & $A_1$ & 8.060\\ 14 & $A_1,21$ & $2A_1$ & 8.055\\ 15 & $A_1,3$ & $A_2+A_1$ & 8.057\\ 25 & $A_5,111$ & $A_5$ & 8.069\\ 26 & $A_5,21$ & $A_5+A_1$ & 8.103\\ 27 & $A_5,3$ & $E_7(a_5)$ & 8.072\\ 28 & $3A_1,111$ & $3A_1$ & 8.078\\ 29 & $3A_1,21$ & $4A_1$ & 8.113\\ 30 & $3A_1,3$ & $A_2+3A_1$ & 8.112\\ 31 & $D_5,111$ & $D_5$ & 8.010\\ 32 & $D_5,21$ & $D_5+A_1$ & 8.071\\ 33 & $D_5,3$ & $D_5+A_2$ & 8.025\\ 34 & $2A_1,111$ & $2A_1$ & 8.055\\ 35 & $2A_1,21$ & $3A_1$ & 8.078\\ 36 & $2A_1,3$ & $A_2+2A_1$ & 8.061\\ 37 & $D_4,111$ & $D_4$ & 8.014\\ 38 & $D_4,21$ & $D_4+A_1$ & 8.105\\ 39 & $D_4,3$ & $D_4+A_2$ & 8.028\\ 40 & $2A_2,111$ & $2A_2$ & 8.053\\ 41 & $2A_2,21$ & $2A_2+A_1$ & 8.088\\ 42 & $2A_2,3$ & $D_4(a_1)$ & 8.040\\ 43 & $E_6(a_3),111$ & $E_6(a_3)$ & 8.023\\ 44 & $E_6(a_3),21$ & $E_6(a_3)+A_1$ & 8.087\\ 45 & $E_6(a_3),3$ & $E_8(a_7)$ & 8.027\\ 46 & $A_2,111$ & $A_2$ & 8.033\\ 47 & $A_2,21$ & $A_2+A_1$ & 8.057\\ 48 & $A_2,3$ & $2A_2$ & 8.053\\ 49 & $A_3+A_1,111$ & $A_3+A_1$ & 8.084\\ 50 & $A_3+A_1,21$ & $A_3+2A_1$ & 8.111\\ 51 & $A_3+A_1,3$ & $A_3+A_2+A_1$ & 8.106\\ 52 & $D_4(a_1),111$ & $D_4(a_1)$ & 8.036\\ 53 & $D_4(a_1),21$ & $D_4(a_1)+A_1$ & 8.059\\ 54 & $D_4(a_1),3$ & $D_4(a_1)+A_2$ & 8.050\\ 58 & $A_4+A_1,111$ & $A_4+A_1$ & 8.041\\ 59 & $A_4+A_1,21$ & $A_4+2A_1$ & 8.051\\ 60 & $A_4+A_1,3$ & $A_4+A_2+A_1$ & 8.049\\ 61 & $A_2+2A_1,111$ & $A_2+2A_1$ & 8.061\\ 62 & $A_2+2A_1,21$ & $A_2+3A_1$ & 8.112\\ 63 & $A_2+2A_1,3$ & $2A_2+2A_1$ & 8.108\\ 64 & $D_5(a_1),111$ & $D_5(a_1)$ & 8.015\\ 65 & $D_5(a_1),21$ & $D_5(a_1)+A_1$ & 8.037\\ 66 & $D_5(a_1),3$ & $D_5(a_1)+A_2$ & 8.099\\ 67 & $A_2+A_1,111$ & $A_2+A_1$ & 8.057\\ 68 & $A_2+A_1,21$ & $A_2+2A_1$ & 8.061\\ 69 & $A_2+A_1,3$ & $2A_2+A_1$ & 8.088\\ 70 & $A_4,111$ & $A_4$ & 8.031\\ 71 & $A_4,21$ & $A_4+A_1$ & 8.041\\ 72 & $A_4,3$ & $A_4+A_2$ & 8.046\\ 73 & $A_3,111$ & $A_3$ & 8.034\\ 74 & $A_3,21$ & $A_3+A_1$ & 8.084\\ 75 & $A_3,3$ & $A_3+A_2$ & 8.052\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 5, $M=A_8$: all 30 trivial-local-system generators.}\label{app:witness-8-5}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $111111111$ & $1$ & 8.054\\ 2 & $21111111$ & $A_1$ & 8.060\\ 3 & $2211111$ & $2A_1$ & 8.055\\ 4 & $222111$ & $3A_1$ & 8.078\\ 5 & $22221$ & $4A_1$ & 8.113\\ 6 & $3111111$ & $A_2$ & 8.033\\ 7 & $321111$ & $A_2+A_1$ & 8.057\\ 8 & $32211$ & $A_2+2A_1$ & 8.061\\ 9 & $3222$ & $A_2+3A_1$ & 8.112\\ 10 & $33111$ & $2A_2$ & 8.053\\ 11 & $3321$ & $2A_2+A_1$ & 8.088\\ 12 & $333$ & $D_4(a_1)$ & 8.040\\ 13 & $411111$ & $A_3$ & 8.034\\ 14 & $42111$ & $A_3+A_1$ & 8.084\\ 15 & $4221$ & $A_3+2A_1$ & 8.111\\ 16 & $4311$ & $A_3+A_2$ & 8.052\\ 17 & $432$ & $A_3+A_2+A_1$ & 8.106\\ 18 & $441$ & $2A_3$ & 8.100\\ 19 & $51111$ & $A_4$ & 8.031\\ 20 & $5211$ & $A_4+A_1$ & 8.041\\ 21 & $522$ & $A_4+2A_1$ & 8.051\\ 22 & $531$ & $A_4+A_2$ & 8.046\\ 23 & $54$ & $A_4+A_3$ & 8.102\\ 24 & $6111$ & $A_5$ & 8.069\\ 25 & $621$ & $A_5+A_1$ & 8.103\\ 26 & $63$ & $E_7(a_5)$ & 8.072\\ 27 & $711$ & $A_6$ & 8.044\\ 28 & $72$ & $A_6+A_1$ & 8.048\\ 29 & $81$ & $A_7$ & 8.070\\ 30 & $9$ & $E_8(a_6)$ & 8.039\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 6, $M=D_5+A_3$: all 80 trivial-local-system generators.}\label{app:witness-8-6}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $222211,1111$ & $2A_1$ & 8.055\\ 2 & $222211,211$ & $3A_1$ & 8.078\\ 3 & $222211,22$ & $4A_1$ & 8.113\\ 4 & $222211,31$ & $A_2+2A_1$ & 8.061\\ 5 & $222211,4$ & $A_3+2A_1$ & 8.111\\ 6 & $22111111,1111$ & $A_1$ & 8.060\\ 7 & $22111111,211$ & $2A_1$ & 8.055\\ 8 & $22111111,22$ & $3A_1$ & 8.078\\ 9 & $22111111,31$ & $A_2+A_1$ & 8.057\\ 10 & $22111111,4$ & $A_3+A_1$ & 8.084\\ 11 & $1111111111,1111$ & $1$ & 8.054\\ 12 & $1111111111,211$ & $A_1$ & 8.060\\ 13 & $1111111111,22$ & $2A_1$ & 8.055\\ 14 & $1111111111,31$ & $A_2$ & 8.033\\ 15 & $1111111111,4$ & $A_3$ & 8.034\\ 21 & $3322,1111$ & $A_2+A_1$ & 8.057\\ 22 & $3322,211$ & $A_2+2A_1$ & 8.061\\ 23 & $3322,22$ & $A_2+3A_1$ & 8.112\\ 24 & $3322,31$ & $2A_2+A_1$ & 8.088\\ 25 & $3322,4$ & $A_3+A_2+A_1$ & 8.106\\ 26 & $331111,1111$ & $A_2$ & 8.033\\ 27 & $331111,211$ & $A_2+A_1$ & 8.057\\ 28 & $331111,22$ & $A_2+2A_1$ & 8.061\\ 29 & $331111,31$ & $2A_2$ & 8.053\\ 30 & $331111,4$ & $A_3+A_2$ & 8.052\\ 31 & $31111111,1111$ & $2A_1$ & 8.055\\ 32 & $31111111,211$ & $3A_1$ & 8.078\\ 33 & $31111111,22$ & $A_2$ & 8.033\\ 34 & $31111111,31$ & $A_2+2A_1$ & 8.061\\ 35 & $31111111,4$ & $D_4(a_1)$ & 8.036\\ 36 & $3331,1111$ & $A_2+2A_1$ & 8.061\\ 37 & $3331,211$ & $A_2+3A_1$ & 8.112\\ 38 & $3331,22$ & $2A_2$ & 8.053\\ 39 & $3331,31$ & $2A_2+2A_1$ & 8.108\\ 40 & $3331,4$ & $D_4(a_1)+A_2$ & 8.050\\ 41 & $4411,1111$ & $A_3$ & 8.034\\ 42 & $4411,211$ & $A_3+A_1$ & 8.084\\ 43 & $4411,22$ & $A_3+2A_1$ & 8.111\\ 44 & $4411,31$ & $A_3+A_2$ & 8.052\\ 45 & $4411,4$ & $2A_3$ & 8.100\\ 46 & $322111,1111$ & $3A_1$ & 8.078\\ 47 & $322111,211$ & $4A_1$ & 8.113\\ 48 & $322111,22$ & $A_2+A_1$ & 8.056\\ 49 & $322111,31$ & $A_2+3A_1$ & 8.112\\ 50 & $322111,4$ & $D_4(a_1)+A_1$ & 8.058\\ 56 & $5311,1111$ & $D_4(a_1)$ & 8.036\\ 57 & $5311,211$ & $D_4(a_1)+A_1$ & 8.059\\ 58 & $5311,22$ & $A_3+A_2$ & 8.052\\ 59 & $5311,31$ & $D_4(a_1)+A_2$ & 8.050\\ 60 & $5311,4$ & $A_4+2A_1$ & 8.114\\ 61 & $511111,1111$ & $A_3$ & 8.034\\ 62 & $511111,211$ & $A_3+A_1$ & 8.084\\ 63 & $511111,22$ & $D_4(a_1)$ & 8.036\\ 64 & $511111,31$ & $A_3+A_2$ & 8.052\\ 65 & $511111,4$ & $A_4$ & 8.031\\ 66 & $55,1111$ & $A_4$ & 8.031\\ 67 & $55,211$ & $A_4+A_1$ & 8.041\\ 68 & $55,22$ & $A_4+2A_1$ & 8.051\\ 69 & $55,31$ & $A_4+A_2$ & 8.046\\ 70 & $55,4$ & $A_4+A_3$ & 8.102\\ 71 & $5221,1111$ & $A_3+A_1$ & 8.084\\ 72 & $5221,211$ & $A_3+2A_1$ & 8.111\\ 73 & $5221,22$ & $D_4(a_1)+A_1$ & 8.058\\ 74 & $5221,31$ & $A_3+A_2+A_1$ & 8.106\\ 75 & $5221,4$ & $A_4+A_1$ & 8.082\\ 76 & $73,1111$ & $D_5(a_1)$ & 8.015\\ 77 & $73,211$ & $D_5(a_1)+A_1$ & 8.037\\ 78 & $73,22$ & $D_4+A_2$ & 8.028\\ 79 & $73,31$ & $D_5(a_1)+A_2$ & 8.099\\ 80 & $73,4$ & $E_8(a_7)$ & 8.045\\ 81 & $7111,1111$ & $D_4$ & 8.014\\ 82 & $7111,211$ & $D_4+A_1$ & 8.105\\ 83 & $7111,22$ & $D_5(a_1)$ & 8.035\\ 84 & $7111,31$ & $D_4+A_2$ & 8.028\\ 85 & $7111,4$ & $D_6(a_2)$ & 8.097\\ 86 & $91,1111$ & $D_5$ & 8.010\\ 87 & $91,211$ & $D_5+A_1$ & 8.071\\ 88 & $91,22$ & $D_6(a_1)$ & 8.013\\ 89 & $91,31$ & $D_5+A_2$ & 8.025\\ 90 & $91,4$ & $D_7(a_2)$ & 8.047\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 7, $M=A_7+A_1$: all 44 trivial-local-system generators.}\label{app:witness-8-7}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $11,11111111$ & $1$ & 8.054\\ 2 & $11,2111111$ & $A_1$ & 8.060\\ 3 & $11,221111$ & $2A_1$ & 8.055\\ 4 & $11,22211$ & $3A_1$ & 8.078\\ 5 & $11,2222$ & $A_2$ & 8.033\\ 6 & $11,311111$ & $A_2$ & 8.033\\ 7 & $11,32111$ & $A_2+A_1$ & 8.057\\ 8 & $11,3221$ & $A_2+2A_1$ & 8.061\\ 9 & $11,3311$ & $2A_2$ & 8.053\\ 10 & $11,332$ & $2A_2+A_1$ & 8.088\\ 11 & $11,41111$ & $A_3$ & 8.034\\ 12 & $11,4211$ & $A_3+A_1$ & 8.084\\ 13 & $11,422$ & $D_4(a_1)$ & 8.036\\ 14 & $11,431$ & $A_3+A_2$ & 8.052\\ 15 & $11,44$ & $A_4$ & 8.031\\ 16 & $11,5111$ & $A_4$ & 8.031\\ 17 & $11,521$ & $A_4+A_1$ & 8.041\\ 18 & $11,53$ & $A_4+A_2$ & 8.046\\ 19 & $11,611$ & $A_5$ & 8.069\\ 20 & $11,62$ & $E_6(a_3)$ & 8.023\\ 21 & $11,71$ & $A_6$ & 8.044\\ 22 & $11,8$ & $E_6(a_1)$ & 8.021\\ 23 & $2,11111111$ & $A_1$ & 8.060\\ 24 & $2,2111111$ & $2A_1$ & 8.055\\ 25 & $2,221111$ & $3A_1$ & 8.078\\ 26 & $2,22211$ & $4A_1$ & 8.113\\ 27 & $2,2222$ & $A_2+A_1$ & 8.056\\ 28 & $2,311111$ & $A_2+A_1$ & 8.057\\ 29 & $2,32111$ & $A_2+2A_1$ & 8.061\\ 30 & $2,3221$ & $A_2+3A_1$ & 8.112\\ 31 & $2,3311$ & $2A_2+A_1$ & 8.088\\ 32 & $2,332$ & $2A_2+2A_1$ & 8.108\\ 33 & $2,41111$ & $A_3+A_1$ & 8.084\\ 34 & $2,4211$ & $A_3+2A_1$ & 8.111\\ 35 & $2,422$ & $D_4(a_1)+A_1$ & 8.058\\ 36 & $2,431$ & $A_3+A_2+A_1$ & 8.106\\ 37 & $2,44$ & $A_4+A_1$ & 8.082\\ 38 & $2,5111$ & $A_4+A_1$ & 8.041\\ 39 & $2,521$ & $A_4+2A_1$ & 8.051\\ 40 & $2,53$ & $A_4+A_2+A_1$ & 8.049\\ 41 & $2,611$ & $A_5+A_1$ & 8.103\\ 42 & $2,62$ & $E_6(a_3)+A_1$ & 8.087\\ 43 & $2,71$ & $A_6+A_1$ & 8.048\\ 44 & $2,8$ & $E_6(a_1)+A_1$ & 8.077\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 8, $M=2A_4$: all 49 trivial-local-system generators.}\label{app:witness-8-8}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $11111,11111$ & $1$ & 8.054\\ 2 & $11111,2111$ & $A_1$ & 8.060\\ 3 & $11111,221$ & $2A_1$ & 8.055\\ 4 & $11111,311$ & $A_2$ & 8.079\\ 5 & $11111,32$ & $A_2+A_1$ & 8.057\\ 6 & $11111,41$ & $A_3$ & 8.081\\ 7 & $11111,5$ & $A_4$ & 8.076\\ 8 & $2111,11111$ & $A_1$ & 8.060\\ 9 & $2111,2111$ & $2A_1$ & 8.055\\ 10 & $2111,221$ & $3A_1$ & 8.078\\ 11 & $2111,311$ & $A_2+A_1$ & 8.057\\ 12 & $2111,32$ & $A_2+2A_1$ & 8.061\\ 13 & $2111,41$ & $A_3+A_1$ & 8.084\\ 14 & $2111,5$ & $A_4+A_1$ & 8.090\\ 15 & $221,11111$ & $2A_1$ & 8.055\\ 16 & $221,2111$ & $3A_1$ & 8.078\\ 17 & $221,221$ & $4A_1$ & 8.113\\ 18 & $221,311$ & $A_2+2A_1$ & 8.061\\ 19 & $221,32$ & $A_2+3A_1$ & 8.112\\ 20 & $221,41$ & $A_3+2A_1$ & 8.111\\ 21 & $221,5$ & $A_4+2A_1$ & 8.051\\ 22 & $311,11111$ & $A_2$ & 8.079\\ 23 & $311,2111$ & $A_2+A_1$ & 8.057\\ 24 & $311,221$ & $A_2+2A_1$ & 8.061\\ 25 & $311,311$ & $2A_2$ & 8.053\\ 26 & $311,32$ & $2A_2+A_1$ & 8.088\\ 27 & $311,41$ & $A_3+A_2$ & 8.052\\ 28 & $311,5$ & $A_4+A_2$ & 8.085\\ 29 & $32,11111$ & $A_2+A_1$ & 8.057\\ 30 & $32,2111$ & $A_2+2A_1$ & 8.061\\ 31 & $32,221$ & $A_2+3A_1$ & 8.112\\ 32 & $32,311$ & $2A_2+A_1$ & 8.088\\ 33 & $32,32$ & $2A_2+2A_1$ & 8.108\\ 34 & $32,41$ & $A_3+A_2+A_1$ & 8.106\\ 35 & $32,5$ & $A_4+A_2+A_1$ & 8.107\\ 36 & $41,11111$ & $A_3$ & 8.081\\ 37 & $41,2111$ & $A_3+A_1$ & 8.084\\ 38 & $41,221$ & $A_3+2A_1$ & 8.111\\ 39 & $41,311$ & $A_3+A_2$ & 8.052\\ 40 & $41,32$ & $A_3+A_2+A_1$ & 8.106\\ 41 & $41,41$ & $2A_3$ & 8.100\\ 42 & $41,5$ & $A_4+A_3$ & 8.102\\ 43 & $5,11111$ & $A_4$ & 8.076\\ 44 & $5,2111$ & $A_4+A_1$ & 8.090\\ 45 & $5,221$ & $A_4+2A_1$ & 8.051\\ 46 & $5,311$ & $A_4+A_2$ & 8.085\\ 47 & $5,32$ & $A_4+A_2+A_1$ & 8.107\\ 48 & $5,41$ & $A_4+A_3$ & 8.102\\ 49 & $5,5$ & $E_8(a_7)$ & 8.098\\ \end{longtable} \endgroup \begingroup\small \setlength{\tabcolsep}{4pt} \begin{longtable}{r p{4.3cm} p{4.6cm} l} \caption{$E_8$, vertex 9, $M=A_5+A_2+A_1$: all 66 trivial-local-system generators.}\label{app:witness-8-9}\\ \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endfirsthead \toprule Row & Local nilpotent orbit & Ambient orbit & Source ID\\\midrule \endhead \bottomrule\endfoot 1 & $111,11,111111$ & $1$ & 8.054\\ 2 & $111,11,21111$ & $A_1$ & 8.060\\ 3 & $111,11,2211$ & $2A_1$ & 8.055\\ 4 & $111,11,222$ & $3A_1$ & 8.078\\ 5 & $111,11,3111$ & $A_2$ & 8.079\\ 6 & $111,11,321$ & $A_2+A_1$ & 8.057\\ 7 & $111,11,33$ & $2A_2$ & 8.053\\ 8 & $111,11,411$ & $A_3$ & 8.081\\ 9 & $111,11,42$ & $A_3+A_1$ & 8.084\\ 10 & $111,11,51$ & $A_4$ & 8.076\\ 11 & $111,11,6$ & $A_5$ & 8.101\\ 12 & $111,2,111111$ & $A_1$ & 8.060\\ 13 & $111,2,21111$ & $2A_1$ & 8.055\\ 14 & $111,2,2211$ & $3A_1$ & 8.078\\ 15 & $111,2,222$ & $A_2$ & 8.079\\ 16 & $111,2,3111$ & $A_2+A_1$ & 8.057\\ 17 & $111,2,321$ & $A_2+2A_1$ & 8.061\\ 18 & $111,2,33$ & $2A_2+A_1$ & 8.088\\ 19 & $111,2,411$ & $A_3+A_1$ & 8.084\\ 20 & $111,2,42$ & $D_4(a_1)$ & 8.083\\ 21 & $111,2,51$ & $A_4+A_1$ & 8.090\\ 22 & $111,2,6$ & $E_6(a_3)$ & 8.094\\ 23 & $21,11,111111$ & $A_1$ & 8.060\\ 24 & $21,11,21111$ & $2A_1$ & 8.055\\ 25 & $21,11,2211$ & $3A_1$ & 8.078\\ 26 & $21,11,222$ & $4A_1$ & 8.113\\ 27 & $21,11,3111$ & $A_2+A_1$ & 8.057\\ 28 & $21,11,321$ & $A_2+2A_1$ & 8.061\\ 29 & $21,11,33$ & $2A_2+A_1$ & 8.088\\ 30 & $21,11,411$ & $A_3+A_1$ & 8.084\\ 31 & $21,11,42$ & $A_3+2A_1$ & 8.111\\ 32 & $21,11,51$ & $A_4+A_1$ & 8.090\\ 33 & $21,11,6$ & $A_5+A_1$ & 8.103\\ 34 & $21,2,111111$ & $2A_1$ & 8.055\\ 35 & $21,2,21111$ & $3A_1$ & 8.078\\ 36 & $21,2,2211$ & $4A_1$ & 8.113\\ 37 & $21,2,222$ & $A_2+A_1$ & 8.056\\ 38 & $21,2,3111$ & $A_2+2A_1$ & 8.061\\ 39 & $21,2,321$ & $A_2+3A_1$ & 8.112\\ 40 & $21,2,33$ & $2A_2+2A_1$ & 8.108\\ 41 & $21,2,411$ & $A_3+2A_1$ & 8.111\\ 42 & $21,2,42$ & $D_4(a_1)+A_1$ & 8.058\\ 43 & $21,2,51$ & $A_4+2A_1$ & 8.051\\ 44 & $21,2,6$ & $E_6(a_3)+A_1$ & 8.092\\ 45 & $3,11,111111$ & $A_2$ & 8.079\\ 46 & $3,11,21111$ & $A_2+A_1$ & 8.057\\ 47 & $3,11,2211$ & $A_2+2A_1$ & 8.061\\ 48 & $3,11,222$ & $A_2+3A_1$ & 8.112\\ 49 & $3,11,3111$ & $2A_2$ & 8.053\\ 50 & $3,11,321$ & $2A_2+A_1$ & 8.088\\ 51 & $3,11,33$ & $D_4(a_1)$ & 8.083\\ 52 & $3,11,411$ & $A_3+A_2$ & 8.052\\ 53 & $3,11,42$ & $A_3+A_2+A_1$ & 8.106\\ 54 & $3,11,51$ & $A_4+A_2$ & 8.085\\ 55 & $3,11,6$ & $E_7(a_5)$ & 8.104\\ 56 & $3,2,111111$ & $A_2+A_1$ & 8.057\\ 57 & $3,2,21111$ & $A_2+2A_1$ & 8.061\\ 58 & $3,2,2211$ & $A_2+3A_1$ & 8.112\\ 59 & $3,2,222$ & $2A_2$ & 8.053\\ 60 & $3,2,3111$ & $2A_2+A_1$ & 8.088\\ 61 & $3,2,321$ & $2A_2+2A_1$ & 8.108\\ 62 & $3,2,33$ & $D_4(a_1)+A_1$ & 8.G\\ 63 & $3,2,411$ & $A_3+A_2+A_1$ & 8.106\\ 64 & $3,2,42$ & $D_4(a_1)+A_2$ & 8.109\\ 65 & $3,2,51$ & $A_4+A_2+A_1$ & 8.107\\ 66 & $3,2,6$ & $E_8(a_7)$ & 8.093\\ \end{longtable} \endgroup % End sections/exceptional_witnesses_8.tex % End sections/certificates.tex % Begin sections/bibliography.tex \begin{thebibliography}{99} \bibitem{Abuaf} R.~Abuaf, \emph{On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations}, Ann. 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