\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern,amsmath,amssymb,mathtools,microtype,needspace} \usepackage[margin=1in]{geometry} \usepackage{xcolor} \usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} \hypersetup{pdftitle={Terminal symplectic singularities for parabolic basic affine spaces},pdfauthor={}} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\C}{\mathbb C} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\A}{\mathbb A} \newcommand{\Gm}{\mathbb G_m} \newcommand{\PP}{\mathbb P} \newcommand{\OO}{\mathcal O} \newcommand{\Aff}{\overline{T^*(G/U_P)}} \newcommand{\g}{\mathfrak g} \newcommand{\p}{\mathfrak p} \newcommand{\lalg}{\mathfrak l} \newcommand{\talg}{\mathfrak t} \newcommand{\uP}{\mathfrak u_P} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\Lie}{Lie} \DeclareMathOperator{\divisor}{div} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\PD}{PD} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\rk}{rk} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Exc}{Exc} \DeclareMathOperator{\Sym}{Sym} \title[Parabolic basic affine spaces]{Terminal symplectic singularities for parabolic basic affine spaces} \date{} \begin{document} \maketitle \section{Introduction} Let $G$ be a simply connected semisimple group over $\C$, let $P\subseteq G$ be a parabolic subgroup, and let $U_P$ be its unipotent radical. We write \[ A_P:=\Gamma(T^*(G/U_P),\OO),\qquad \Aff:=\Spec A_P. \] The algebra $A_P$ is finitely generated by \cite[Corollary 2.1.7]{Gin25}. Thus $\Aff$ is an affine variety. It is normal: a rational function integral over $A_P$ belongs to every local ring of the smooth variety $T^*(G/U_P)$, and hence belongs to $A_P$. Recall that a normal variety has \emph{symplectic singularities} if its smooth locus admits a symplectic form whose pullback extends to a regular two-form on a resolution of singularities. A normal variety is \emph{$\Q$-factorial} if every Weil divisor has a positive integer multiple that is Cartier. If $V$ is normal and $K_V$ is $\Q$-Cartier, the \emph{discrepancies} of a resolution $f\colon W\to V$ are the coefficients $a_E$ in \[ K_W=f^*K_V+\sum_{E\text{ exceptional}}a_EE. \] The variety $V$ has \emph{terminal singularities} if all these coefficients are positive on every resolution. The goal of this paper is to prove the following theorem. \begin{theorem}\label{thm:main} The variety $\Aff$ has $\Q$-factorial terminal symplectic singularities. Moreover, $A_P$ is a unique factorization domain. \end{theorem} When $P$ is a Borel subgroup, Theorem~\ref{thm:main} recovers the simply connected case of \cite[Theorems 1.1 and 1.2]{Gan24}. The proof of factoriality in \cite[Section 3.1]{Gan24} applies to $U_P$ without change; we recall it in Proposition~\ref{prop:factorial}. The remaining assertions are proved separately in Theorems~\ref{thm:symplectic} and~\ref{thm:terminal}. \subsection{Outline of the proof} Choose a Borel subgroup $B\subseteq P$ and put $\g=\Lie(G)$ and $\p=\Lie(P)$. The criterion of Fu--Liu \cite[Theorem 1.6(1.6.2)]{FL26} reduces the symplectic-singularity assertion to a positivity statement for the projective bundle associated to $G\times^B\p$. We prove that statement in Proposition~\ref{prop:fano} by constructing an effective rational boundary. The main step, Lemma~\ref{lem:levi-boundary}, uses the Frobenius splitting of a flag variety to obtain discrepancy inequalities for finitely many fixed polynomials on a Levi subalgebra. Its proof transfers these inequalities to characteristic zero on a common log resolution. Lemma~\ref{lem:global-boundary} controls the centers at which equality occurs, and Proposition~\ref{prop:fano} uses this control to perturb the boundary. To prove terminality, we first show in Proposition~\ref{prop:rigid} that every first-order Poisson deformation of $\Aff$ is trivial, using the left $G$-action and the right action of a Levi subgroup. Lemma~\ref{lem:namikawa} deduces from Namikawa's deformation theory that the singular locus then has codimension at least four. Finally, \cite[Corollary 1]{Nam01} identifies this condition with terminality for symplectic singularities. \begin{remark}\label{rem:nonconical} A \emph{conical grading} on a finitely generated $\C$-algebra $A$ means a grading $A=\bigoplus_{d\geq0}A_d$ with $A_0=\C$. Such a grading need not exist for $A_P$. Indeed, if $P=G$, then $U_P=\{1\}$ and \[ \Aff=T^*G\cong G\times\g^*. \] For $G=\mathrm{SL}_2$, scalar multiplication contracts the cotangent fibers. Polar decomposition gives the deformation retraction $g\mapsto g(g^*g)^{-t/2}$, for $0\leq t\leq1$, onto $\mathrm{SU}(2)$. The description \[ \mathrm{SU}(2)=\left\{\begin{pmatrix}a&b\\-\bar b&\bar a\end{pmatrix}: |a|^2+|b|^2=1\right\} \] therefore gives homotopy equivalences \[ T^*\mathrm{SL}_2(\C)\simeq\mathrm{SL}_2(\C) \simeq\mathrm{SU}(2)\cong S^3. \] Consequently, \[ H^i(T^*\mathrm{SL}_2(\C),\C)= \begin{cases}\C,&i=0,3,\\0,&\text{otherwise}.\end{cases} \] A conical grading would define a morphism $\A^1\times\Spec A\to\Spec A$ by $a_d\mapsto t^da_d$. At $t=0$ this morphism is constant, because $A_0=\C$, and at $t=1$ it is the identity. Restricting $t$ to $[0,1]$ would therefore contract the complex analytic space, contradicting the displayed nonvanishing of $H^3$. Thus $\Gamma(T^*\mathrm{SL}_2,\OO)$ admits no conical grading. \end{remark} \subsection{Notation and conventions} All varieties are over $\C$, except during the Frobenius-splitting arguments in Lemmas~\ref{lem:flag-splitting} and~\ref{lem:levi-boundary}. Fix a maximal torus $T\subseteq B$ and the Levi subgroup $L\subseteq P$ containing $T$. Put $\uP=\Lie(U_P)$ and set \[ B_L=B\cap L,\qquad \talg=\Lie(T),\qquad \lalg=\Lie(L),\qquad \p=\lalg\oplus\uP. \] Let $\Phi_G^+$ denote the positive roots of $G$ determined by $B$. For reduction modulo a prime, we choose a compatible split rational form and a Chevalley integral model. For a representation $V$ of a subgroup $H\subseteq G$, the associated bundle $G\times^H V$ is the quotient by $(g,v)h=(gh,h^{-1}v)$. Our convention for projectivization is \[ \PP_S(\mathcal V)=\operatorname{Proj}_S\Sym\mathcal V. \] Thus $\PP_S(\mathcal V^*)$ parametrizes lines in the fibers of $\mathcal V$. We use additive notation for rational divisor classes and their associated rational line bundles. If $G$ is trivial, its cotangent bundle and its affinization are a point, so Theorem~\ref{thm:main} is immediate. We therefore assume $\dim G>0$ from now on. \section{Finite generation and factoriality} The fiber of $T^*(G/U_P)$ at the identity coset is $(\g/\uP)^*=\uP^\perp$. The Killing form pairs opposite root spaces and is nondegenerate on $\talg$, so it identifies this annihilator with $\p$. Equivariance then identifies the cotangent bundle as \begin{equation}\label{eq:cotangent-model} M:=T^*(G/U_P)\cong G\times^{U_P}\p. \end{equation} In particular, \[ A_P=\C[G\times\p]^{U_P}. \] We use the finite generation of this algebra from \cite[Corollary 2.1.7]{Gin25}. \begin{proposition}\label{prop:factorial} The algebra $A_P$ is a unique factorization domain. In particular, $\Aff$ is $\Q$-factorial. \end{proposition} \begin{proof} The homogeneous space $G/U_P$ is quasi-affine by \cite[Proposition 14.26]{Mil22}. An affine morphism over a quasi-affine variety has quasi-affine source, by \cite[Tag 01VG, Lemma 29.38.7(2)]{Stacks}; applying this to the vector bundle shows that $M$ is quasi-affine. The canonical map \[ j\colon M\longrightarrow\Aff \] is therefore an open immersion, and its complement has codimension at least two, by \cite[Theorem 2.5]{FL26}; the codimension statement is also \cite[Lemma 3.4]{Gan24}. Removing a subset of codimension at least two does not change Weil divisors or their principal relations, while smoothness identifies Weil and Cartier divisor classes. Consequently, \begin{equation}\label{eq:class-pic} \Cl(\Aff)\cong\Cl(M)=\Pic(M)\cong\Pic(G/U_P), \end{equation} where the last isomorphism is invariance of the Picard group under passage to a vector bundle on a smooth variety, as used in \cite[proof of Proposition 3.3]{Gan24}. The $U_P$-torsor $G\to G/U_P$ is Zariski locally trivial by \cite[Proposition 16.55]{Mil22}. Its Picard sequence \cite[Theorem 18.32]{Mil22} therefore gives \[ \operatorname{Hom}(U_P,\Gm)\longrightarrow\Pic(G/U_P) \longrightarrow\Pic(G)\longrightarrow\Pic(U_P). \] The first group vanishes by \cite[Corollary 14.18]{Mil22}, and $\Pic(G)=0$ by \cite[Corollary 18.24]{Mil22}. Exactness therefore gives $\Pic(G/U_P)=0$, so \eqref{eq:class-pic} gives $\Cl(\Aff)=0$. A Noetherian normal domain with trivial divisor class group is a unique factorization domain, as used in \cite[Corollary 3.5]{Gan24}. In particular, every Weil divisor on $\Aff$ is principal, which implies $\Q$-factoriality. \end{proof} \begin{lemma}\label{lem:smooth-form} The cotangent symplectic form on $M$ extends uniquely to a symplectic form on the smooth locus of $\Aff$. \end{lemma} \begin{proof} By the proof of Proposition~\ref{prop:factorial}, the complement of $M$ has codimension at least two. A section of a locally free sheaf on a smooth variety extends uniquely across such a subset: after trivializing the sheaf, this is the extension property for regular functions on a normal variety. Apply this to the sheaf of two-forms on the smooth locus to extend the cotangent form to $\omega$. The form $d\omega$ vanishes on the dense open set $M$, hence vanishes everywhere. If $\dim M=2d$, the section $\omega^{\wedge d}$ of the canonical line bundle is nonzero on $M$. A nonzero section of a line bundle on a smooth integral variety has either an empty zero locus or a divisor as its zero locus. The codimension statement therefore forces $\omega^{\wedge d}$ to be nowhere zero, so $\omega$ is symplectic. \end{proof} \section{Symplectic singularities} We first prove the symplectic-singularity assertion independently of terminality. \begin{theorem}\label{thm:symplectic} The canonical symplectic form on $T^*(G/U_P)$ extends to the smooth locus of $\Aff$ and makes $\Aff$ a variety with symplectic singularities. \end{theorem} We recall the discrepancy terminology needed for the proof. If $V$ is smooth and $D$ is an effective rational divisor, the \emph{log discrepancy} of a prime divisor $E$ on a smooth birational model $f\colon\widetilde V\to V$ is \[ a_E(V,D)=1+\ord_E(K_{\widetilde V}-f^*(K_V+D)). \] The pair $(V,D)$ is \emph{log canonical} if all its log discrepancies are nonnegative, and \emph{Kawamata log terminal}, or \emph{klt}, if they are positive. A \emph{log canonical center} is the image of a divisor with log discrepancy zero. A smooth projective variety $V$ is \emph{of Fano type} if there is an effective rational divisor $\Delta$ such that $(V,\Delta)$ is klt and $-(K_V+\Delta)$ is ample. A \emph{log resolution} of $(V,D)$ is a proper birational morphism $f\colon\widetilde V\to V$ with $\widetilde V$ smooth such that the exceptional divisor together with the support of $f^*D$ has simple normal crossings. We recall three standard facts about log canonical pairs. \begin{lemma}\label{lem:pair-facts} Let $V$ be smooth and let $D$ be an effective rational divisor. \begin{enumerate} \item Log canonicity of $(V,D)$ is preserved and detected by smooth surjective pullback. \item The pair $(V,D)$ has finitely many log canonical centers if it is log canonical. \item If $(V,D)$ is log canonical and $0<\epsilon<1$, then $(V,(1-\epsilon)D)$ is klt. \end{enumerate} \end{lemma} \begin{proof} Let $f\colon\widetilde V\to V$ be a log resolution, and write $K_{\widetilde V}+\widetilde D=f^*(K_V+D)$. The log resolution criterion \cite[Remark 3.2(1)]{HW02} says that $(V,D)$ is log canonical, respectively klt, if and only if every coefficient of $\widetilde D$ is at most one, respectively less than one. Given a smooth surjection $g\colon W\to V$, the base change $\widetilde V\times_V W\to W$ is a log resolution of $(W,g^*D)$. The relative canonical divisor and the boundary pull back under this base change, so the coefficients in the criterion are unchanged. Surjectivity ensures that every component of $\widetilde D$ has nonempty inverse image. Thus the criterion holds after pullback if and only if it holds before pullback, proving (1). Part (2) is \cite[Theorem 9.1(1)]{Fuj11}. Finally, $(V,0)$ is klt by the same log resolution criterion, and $D$ is $\Q$-Cartier because $V$ is smooth. Hence (3) follows from \cite[Corollary 2.35(5)]{KM98}, applied with $\Delta=0$, $\Delta'=D$, and $c=1-\epsilon$ in the notation of that reference. \end{proof} \subsection{The two projective bundles} Define \[ E:=G\times^B\p,\qquad F:=G\times^P\p, \qquad Y:=\PP_{G/B}(E^*),\qquad Z:=\PP_{G/P}(F^*). \] Since $E$ is the pullback of $F$, there is a Cartesian square \[ \begin{array}{ccc} Y&\longrightarrow&G/B\\ \pi\downarrow&&\downarrow\\ Z&\longrightarrow&G/P. \end{array} \] Let $A$ be the class of $\OO_Z(1)$, let $\xi=\pi^*A$, and set $R=-K_{Y/Z}$. Write $\Phi_L^+$ for the positive roots of $L$ determined by $B_L$, and put \[ n=|\Phi_L^+|,\qquad 2\rho_L=\sum_{\alpha\in\Phi_L^+}\alpha, \qquad r=\dim\p. \] \begin{lemma}\label{lem:bundles} The divisor $A$ is globally generated, nef, and big. The divisor $R$ is $\pi$-ample, and \begin{equation}\label{eq:canonical} -K_Z=rA,\qquad -K_Y=r\xi+R,\qquad r>n. \end{equation} Moreover, $R$ is the pullback from $G/B$ of the line bundle associated to the $B$-character $-2\rho_L$. \end{lemma} \begin{proof} The inclusion $F\subseteq(G/P)\times\g$, given by $[g,x]\mapsto(gP,\Ad_gx)$, gives a surjection from a trivial bundle to $F^*$. Thus $A$ is generated by the subspace of sections coming from $\g^*$, and this subspace defines the morphism \begin{equation}\label{eq:q} q\colon Z\longrightarrow\PP(\g^*),\qquad [g,[x]]\longmapsto[\Ad_gx]. \end{equation} Here $\PP(\g^*)$ parametrizes lines in $\g$, and $A=q^*\OO_{\PP(\g^*)}(1)$ by construction. This proves global generation and nefness, since the degree of this pullback on any curve is nonnegative. Call an element of $\g$ \emph{regular semisimple} if it is semisimple and its centralizer is a maximal torus. If $x$ is such an element, a parabolic whose Lie algebra contains $x$ contains $C_G(x)$: apply \cite[Proposition 17.90]{Mil22} to that parabolic to obtain a maximal torus centralizing $x$, and compare dimensions inside $C_G(x)$. There are finitely many parabolics of a given type containing a fixed maximal torus, since their Lie algebras are sums of the toral subalgebra and root spaces. Hence the fibers of $q$ over regular semisimple lines are finite. They are nonempty, since a maximal torus is contained in a parabolic of the prescribed type. As $q$ is proper, its restriction over this open set is finite. Thus $q$ is dominant and generically finite, which proves that $A=q^*\OO(1)$ is big. Opposite Levi root spaces cancel in $\det\p$, so its character is $\sum_{\alpha\in\Phi_G^+\setminus\Phi_L^+}\alpha$. The same character is the determinant of $(\g/\p)^*$, the cotangent fiber of $G/P$. Hence $\det F=K_{G/P}$. If $s\colon Z\to G/P$ is the projection, the relative Euler sequence for the projective bundle gives \[ K_Z=-rA+s^*(K_{G/P}-\det F)=-rA. \] The identity $K_Y=\pi^*K_Z+K_{Y/Z}$ now gives the second equality in \eqref{eq:canonical}; compare \cite[Proposition 5.5]{FL26}. The relative tangent bundle of $G/B\to G/P$ is $G\times^B(\p/\Lie(B))$. Its determinant has character $-2\rho_L$, because the weights of $\p/\Lie(B)$ are exactly the negative Levi roots. Base change in the displayed Cartesian square proves the asserted global description of $R$. On every fiber $L/B_L$, this line bundle is ample: $2\rho_L$ pairs positively with every simple Levi coroot, so the flag-variety ampleness criterion \cite[Exercise 3.1.E.1]{BK05} applies. Over each translate of the opposite big cell of $G/P$, the root-subgroup decomposition trivializes $G\to G/P$ and hence $G/B\to G/P$; see \cite[\S2.1]{BK05}. Thus $\pi$ is locally a product with $L/B_L$, and its relative anticanonical bundle $R$ is $\pi$-ample. Finally, the root decomposition gives \[ r=\dim\lalg+\dim\uP=2n+\rk L+\dim\uP>n. \qedhere \] \end{proof} By \cite[Theorem 1.6(1.6.2)]{FL26}, Theorem~\ref{thm:symplectic} follows once we prove that $Y$ is of Fano type. We will construct a log canonical divisor of class $R+n\xi$ whose log canonical centers meet the open set on which $q$ is finite. \subsection{A boundary on the Levi subalgebra} We first state the Frobenius splitting fact used below. For a variety $\mathcal F$ in characteristic $p>0$, its absolute Frobenius is denoted by $\operatorname{Fr}$. A \emph{Frobenius splitting} is an $\OO_{\mathcal F}$-linear retraction of $\OO_{\mathcal F}\to\operatorname{Fr}_*\OO_{\mathcal F}$, $a\mapsto a^p$. \begin{lemma}[Brion--Kumar]\label{lem:flag-splitting} Let $G_0$ be a simply connected semisimple group over an algebraically closed field $k$ of characteristic $p>0$, and choose a maximal torus $T_0$ in a Borel subgroup $B_0$. Put $\mathcal F=G_0/B_0$, let $\rho_0$ be the half-sum of positive roots, and set \[ \mathcal L(\rho_0)=G_0\times^{B_0}k_{-\rho_0}. \] Choose nonzero sections $s_+$ and $s_-$ of $\mathcal L(\rho_0)$ of highest and lowest weight, respectively. Under the identifications \[ \mathcal L(2\rho_0)\cong\omega_{\mathcal F}^{-1},\qquad H^0(\mathcal F,\omega_{\mathcal F}^{1-p}) \cong\operatorname{Hom}_{\OO_{\mathcal F}} (\operatorname{Fr}_*\OO_{\mathcal F},\OO_{\mathcal F}), \] a nonzero scalar multiple of $(s_+s_-)^{p-1}$ defines a Frobenius splitting. On the opposite big cell, with its usual trivialization of $\mathcal L(\rho_0)$ and suitable normalizations, $s_-=1$ and \[ s_+(u)=\ell_-(uv_+), \] where $v_+$ is a highest-weight vector of the Weyl module of highest weight $\rho_0$, and $\ell_-$ extracts its lowest-weight component. \end{lemma} \begin{proof} The canonical-bundle identification is \cite[equation (2.1.8)]{BK05}, and the Frobenius-duality identification is \cite[Theorem 1.3.8]{BK05}. Theorem 2.3.1 of \cite{BK05} says that the product of the highest- and lowest-weight sections of the Steinberg module defines a splitting up to a nonzero scalar. Equations (4)--(5) of its proof identify those sections with the $(p-1)$st powers of $s_+$ and $s_-$. The evaluation formula in equations (1)--(2) of that proof gives the asserted big-cell expressions: the functional extracting the highest-weight component is constant on $uv_+$ for $u$ in the opposite unipotent subgroup, while the other extremal functional extracts the lowest-weight component. These statements occur on pages 72--74 of \cite{BK05}. \end{proof} The next lemma supplies both the divisor class and the required control of its centers. A polynomial $f$ is called a $B_L$-semi-invariant of character $\chi$ if $f(\Ad_bx)=\chi(b)f(x)$ for all $b\in B_L$. \begin{lemma}\label{lem:levi-boundary} Assume $n>0$. Let $\delta\in\C[\lalg]^L$ be a homogeneous invariant polynomial defined over $\Q$ with nonzero restriction to $\talg$. There are nonzero homogeneous polynomials $f_d\in\Q[\lalg]$, indexed by a finite set of positive degrees, and nonnegative rational numbers $c_d$ such that \begin{equation}\label{eq:boundary-weights} f_d(\Ad_bx)=(-2\rho_L)(b)f_d(x),\qquad \sum_dc_d=1,\qquad \sum_ddc_d=n. \end{equation} For some rational $\eta>0$, the pair \begin{equation}\label{eq:levi-lc} \left(\lalg,\sum_dc_d\divisor(f_d)+\eta\divisor(\delta)\right) \end{equation} is log canonical. \end{lemma} \begin{proof} Let $J$ be the set of simple roots of $L$. Since $G$ is simply connected, its fundamental weights are characters of $T$; take $\lambda$ to be the sum of the fundamental weights indexed by $J$. Its pairing with every simple Levi coroot is one, so its restriction to the derived torus is $\rho_L$. The highest-weight theorem \cite[Theorem 22.2]{Mil22} supplies an irreducible $L$-representation $V_\lambda$, which we choose over the fixed split rational form. Let $v_+$ be a rational highest-weight vector and let $\ell_-$ be a rational functional extracting the lowest-weight component. If $w_{0,L}$ is the longest element of the Levi Weyl group, then $w_{0,L}\lambda-\lambda=-2\rho_L$: the difference $\lambda-\rho_L$ is fixed by that Weyl group and $w_{0,L}\rho_L=-\rho_L$. For $1\leq d\leq d_{\max}:=\dim V_\lambda-1$, set \begin{equation}\label{eq:fd} f_d(x)=\frac{1}{d!}\ell_-(x^dv_+), \end{equation} where $x$ acts through the differential of the representation. Discard any polynomial that is identically zero. For $b\in B_L$, the extremal weight properties give $b^{-1}v_+=\lambda(b)^{-1}v_+$ and $\ell_-\circ b=(w_{0,L}\lambda)(b)\ell_-$. Substitution in \eqref{eq:fd} gives \[ f_d(\Ad_bx)=(w_{0,L}\lambda-\lambda)(b)f_d(x) =(-2\rho_L)(b)f_d(x). \] This proves the first equality in \eqref{eq:boundary-weights}, while \eqref{eq:fd} proves homogeneity of degree $d$. Let $\mathfrak u_L$ be the nilpotent radical of $\Lie(B_L)$, and let $\mathfrak u_L^-$ be the opposite nilpotent subalgebra. Choose root coordinates $z_\alpha$ on $\mathfrak u_L^-$, for $\alpha\in\Phi_L^+$. On this subalgebra, the polynomial \begin{equation}\label{eq:f-exp} f:=\sum_df_d=\ell_-(\exp(x)v_+) \end{equation} is the extremal matrix coefficient on the opposite big cell of $L/B_L$: nilpotence on $V_\lambda$ makes the exponential a sum through degree $d_{\max}$, and the constant term is zero because the two extremal weights are distinct. The line bundle involved restricts to the line bundle of weight $\rho_L$ on the flag variety of the simply connected cover of the derived group; that flag variety is isomorphic to $L/B_L$. Choose the integral Weyl-module form of this representation, so its reductions give the Weyl modules in Lemma~\ref{lem:flag-splitting}. After inverting finitely many integers, the polynomials and the exponential isomorphism $\mathfrak u_L^-\cong U_L^-$ are defined over the resulting localization of $\Z$. Here the characteristic-zero exponential isomorphism is \cite[Proposition 14.32]{Mil22}, and its inverse and the identities defining it spread out after clearing finitely many denominators. Lemma~\ref{lem:flag-splitting} implies that, for every sufficiently large prime $p$, the coefficient \begin{equation}\label{eq:trace-coefficient} \left[\prod_{\alpha\in\Phi_L^+}z_\alpha^{p-1}\right]f^{p-1} \end{equation} is nonzero modulo $p$. Indeed, that lemma identifies the splitting on the opposite big cell with a nonzero scalar multiple of $f^{p-1}$. Under the exponential isomorphism, the canonical-bundle trivialization used for this splitting differs from $\bigwedge_\alpha dz_\alpha$ by a nonzero constant, because both trivializations are nowhere zero on an affine space and its only units are constants. Thus the Frobenius trace of $f^{p-1}$ is a nonzero constant. The trace formula \cite[Theorem 1.3.8]{BK05} identifies its constant term with the coefficient in \eqref{eq:trace-coefficient}, up to taking a $p$th root. This proves the required nonvanishing in exponential coordinates. Expanding $f^{p-1}$, we may choose nonnegative integers $m_d$ for which \begin{equation}\label{eq:md} F_p:=\prod_df_d^{m_d},\qquad \sum_dm_d=p-1, \end{equation} has nonzero coefficient at $\prod_\alpha z_\alpha^{p-1}$ after restriction to $\mathfrak u_L^-$. Since $F_p$ is homogeneous, this also gives \begin{equation}\label{eq:degreeFp} \sum_ddm_d=n(p-1). \end{equation} We next include the invariant polynomial $\delta$. Write \[ \lalg=\mathfrak u_L^-\oplus\talg\oplus\mathfrak u_L, \] with coordinates $z_\alpha$, $h_1,\ldots,h_s$, and $y_\alpha$, respectively, where $s=\dim\talg$, and set $\mathfrak b_L^-=\talg\oplus\mathfrak u_L^-$. Choose a cocharacter pairing negatively with every simple Levi root, by taking a sufficiently divisible negative sum of the fundamental Levi coweights. It fixes $\talg$ and contracts $\mathfrak u_L^-$ as its parameter tends to zero. Invariance of $\delta$ then gives $\delta(h+z)=\delta(h)$, so $\delta|_{\mathfrak b_L^-}$ is the pullback of $\delta|_{\talg}$. Choose a monomial \[ c h_1^{a_1}\cdots h_s^{a_s} \] with nonzero coefficient in $\delta|_{\talg}$. Take $p$ larger than all the $a_i$ and avoiding the numerator and denominator of $c$, and set \[ Q_p=\prod_{\alpha\in\Phi_L^+}y_\alpha^{p-1} \prod_{i=1}^s h_i^{p-1-a_i}. \] The coefficient of \begin{equation}\label{eq:full-monomial} \prod_\alpha z_\alpha^{p-1}\prod_i h_i^{p-1}\prod_\alpha y_\alpha^{p-1} \end{equation} in $Q_pF_p\delta$ is nonzero. Indeed, the $y$-exponents already supplied by $Q_p$ force us to restrict $F_p\delta$ to $\mathfrak b_L^-$. There $\delta$ depends only on $h$. The required $z$-degree is $n(p-1)$, which is the whole degree of $F_p$ by \eqref{eq:degreeFp}. Thus only $F_p|_{\mathfrak u_L^-}$ contributes, and the coefficient is the product of $c$ and the nonzero coefficient chosen in \eqref{eq:md}. The total degree of $Q_pF_p\delta$ is $(2n+s)(p-1)$, by \eqref{eq:degreeFp} and the definition of $Q_p$. For the polynomial ring $S=\overline{\mathbb F}_p[\lalg]$, let $\Tr\colon F_*S\to S$ be the Frobenius trace with respect to the chosen coordinate volume form; here $F_*S$ has scalar action $a\cdot u=a^pu$. By \cite[Theorem 1.3.8]{BK05}, this trace kills a monomial unless every exponent is congruent to $p-1$ modulo $p$, and sends \eqref{eq:full-monomial} to $1$. The degree equality leaves only that monomial, so $\Tr(Q_pF_p\delta)$ is a nonzero constant. Rescale $Q_p$ to make this constant $1$. This gives a splitting for the pair \begin{equation}\label{eq:charp-pair} \left(\lalg_{\overline{\mathbb F}_p}, \frac{1}{p-1}\divisor(F_p\delta)\right). \end{equation} More explicitly, put $H_p=F_p\delta$. The $S$-linear map \[ F_*(H_p^{-1}S)\longrightarrow S,\qquad u\longmapsto\Tr(Q_pH_pu) \] sends $1$ to $1$ and retracts $S\to F_*(H_p^{-1}S)$, $a\mapsto a^p$. Its iterates retract the analogous inclusions for $F^e$ with pole divisor $\frac{p^e-1}{p-1}\divisor(H_p)$. Since this divisor is exactly $(p^e-1)$ times the boundary in \eqref{eq:charp-pair}, the pair is $F$-pure in the sense of \cite[Definition 2.1]{HW02}. It remains to justify the passage to characteristic zero, since the integers $m_d$ depend on $p$. Before choosing $p$, fix a log resolution of \[ \divisor\left(\delta\prod_df_d\right) \] over a number field. Spread the morphism, its inverse over a dense open set, and its finitely many boundary components out over a localization of the ring of integers. After further localization, the morphism remains proper and birational, its source remains smooth, and its total boundary has relative simple normal crossings. Moreover, the divisor equalities for the pullbacks of the finitely many $f_d$, of $\delta$, and for the relative canonical divisor persist on each fiber. These are precisely the resolution and divisor data spread out in \cite[proof of Theorem 3.7]{HW02}. Choose a sufficiently large good residue characteristic $p$, and only then choose the integers $m_d$ in \eqref{eq:md}. Hara--Watanabe's discrepancy theorem in fixed positive characteristic, \cite[Theorem 3.3]{HW02}, applies locally to the $F$-pure pair \eqref{eq:charp-pair}; its ambient space is smooth and its boundary is rational Cartier. It gives log discrepancies at least zero for the exceptional divisors of the reduced resolution, while \cite[Proposition 2.2(4)]{HW02} gives the same inequality for the strict transforms of boundary components. Writing $k_E$ for the relative canonical coefficient and $v_E$ for the component order, each inequality is \begin{equation}\label{eq:spread-discrepancy} 1+k_E-\sum_d\frac{m_d}{p-1}v_E(f_d) -\frac{1}{p-1}v_E(\delta)\geq0. \end{equation} The integers $k_E$, $v_E(f_d)$, and $v_E(\delta)$ in \eqref{eq:spread-discrepancy} were preserved by the preceding spread. Keeping the chosen coefficients $m_d/(p-1)$ and $1/(p-1)$, the same inequality therefore holds on the fixed characteristic-zero resolution. The log resolution criterion \cite[Remark 3.2(1)]{HW02} proves that \eqref{eq:levi-lc} is log canonical for \[ c_d=\frac{m_d}{p-1},\qquad \eta=\frac{1}{p-1}. \] Equations \eqref{eq:md} and \eqref{eq:degreeFp} give the remaining equalities in \eqref{eq:boundary-weights}. \end{proof} \subsection{Descent and perturbation of the boundary} We first fix an invariant polynomial detecting the regular semisimple locus. Let $\delta_G$ be $(-1)^{|\Phi_G^+|}$ times the coefficient of $t^{\rk G}$ in $\det(t-\operatorname{ad}x)$. This definition makes $\delta_G$ homogeneous, invariant, and defined over the fixed rational form; the root decomposition gives \[ \delta_G|_{\talg}=\prod_{\alpha\in\Phi_G^+}\alpha^2. \] For the Jordan decomposition $x=x_s+x_n$, the commuting nilpotent operator $\operatorname{ad}x_n$ does not change the characteristic polynomial of $\operatorname{ad}x_s$. Thus $\delta_G(x)=\delta_G(x_s)$, and the displayed product is nonzero precisely when $x_s$ is regular. In that case $x_n$ belongs to the toral centralizer of $x_s$ and is nilpotent, so $x_n=0$. This proves that the nonvanishing locus of $\delta_G$ is exactly the regular semisimple locus. Choose a cocharacter $\gamma\colon\Gm\to Z(L)^\circ$ with strictly positive weights on $\uP$: a sufficiently divisible sum of the fundamental coweights indexed by the simple roots outside $L$ has these properties. If $P=G$, take $\gamma=0$. Since $\gamma$ fixes $\lalg$ and contracts $\uP$, invariance gives \begin{equation}\label{eq:discriminant-projection} \delta_G(x_l+x_u)=\lim_{t\to0}\delta_G(\Ad_{\gamma(t)}(x_l+x_u)) =\delta_G(x_l). \end{equation} Set $\delta=\delta_G|_{\lalg}$, which is nonzero on $\talg$ by the product formula. Let $Z^{\mathrm{rs}}\subseteq Z$ be the inverse image under $q$ of the regular semisimple lines. \begin{lemma}\label{lem:global-boundary} There is an effective rational divisor $D$ on $Y$ such that \begin{equation}\label{eq:D-class} D\sim_\Q R+n\xi, \end{equation} the pair $(Y,D)$ is log canonical, and every log canonical center of $(Y,D)$ meets $\pi^{-1}(Z^{\mathrm{rs}})$. \end{lemma} \begin{proof} Suppose first that $n>0$, and apply Lemma~\ref{lem:levi-boundary} to the polynomial $\delta$ just constructed. Pull its polynomials $f_d$ back through $\p\to\lalg$. This projection is $B$-equivariant for the quotient $B\to B_L$, so the pulled-back polynomials remain semi-invariants of character $-2\rho_L$. Their semi-invariance and homogeneity imply that their zero divisors descend along the smooth quotient \[ G\times(\p\setminus\{0\})\longrightarrow Y \] to divisors $D_d$ with \[ D_d\sim R+d\xi. \] Indeed, $\OO_Y(-1)$ is the tautological line in the bundle with fiber $\p$. The polynomial $f_d$ gives a degree-$d$ evaluation map to the line of character $-2\rho_L$, which is $\OO_Y(R)$ by Lemma~\ref{lem:bundles}. Thus $f_d$ defines a section of $\OO_Y(d)\otimes\OO_Y(R)$, proving the displayed class. Put $D=\sum_dc_dD_d$. Equation \eqref{eq:boundary-weights} gives \eqref{eq:D-class}. Let $D_\delta$ be the divisor on $Y$ induced by $\delta_G|_{\p}$. Invariance of $\delta_G$ and the definition of $q$ show that it is the pullback under $q\pi$ of the hypersurface defined by $\delta_G$. The map $G\times\p\to\lalg$ is smooth and surjective, since it is a product projection followed by a vector-space projection. Pull back the pair in Lemma~\ref{lem:levi-boundary} along this map and restrict to $G\times(\p\setminus\{0\})$; it remains log canonical by Lemma~\ref{lem:pair-facts}(1) and the definition of log canonicity on an open set. The quotient to $Y$ is a torsor under the smooth group $B\times\Gm$, so smooth descent in Lemma~\ref{lem:pair-facts}(1), together with \eqref{eq:discriminant-projection}, gives \begin{equation}\label{eq:lc-delta} (Y,D+\eta D_\delta)\quad\text{log canonical}. \end{equation} Dropping the effective summand $\eta D_\delta$ in \eqref{eq:lc-delta} proves that $(Y,D)$ is log canonical. If a log canonical center of $(Y,D)$ were contained in $D_\delta$, a prime divisor $E$ defining it would have $\ord_E(D_\delta)>0$. Its log discrepancy for \eqref{eq:lc-delta} would be $0-\eta\ord_E(D_\delta)<0$, a contradiction. Since $Y\setminus\Supp D_\delta=\pi^{-1}(Z^{\mathrm{rs}})$, this proves the assertion about centers. If $n=0$, then $L$ is a torus, $P=B$, $Y=Z$, and $R=0$. Take $D=0$. The smooth pair $(Y,0)$ is klt and has no log canonical centers, so all the assertions hold. \end{proof} \begin{proposition}\label{prop:fano} The variety $Y=\PP_{G/B}(E^*)$ is of Fano type. \end{proposition} \begin{proof} Choose $D$ as in Lemma~\ref{lem:global-boundary} and an ample Cartier divisor $H$ on $Z$. For sufficiently large $m$, the linear system $|mA-H|$ has no base points on $Z^{\mathrm{rs}}$, as we now verify. The coherent sheaf $q_*\OO_Z(-H)(m)$ is globally generated for $m$ sufficiently large by \cite[Tag 01Q3, Proposition 28.27.13(7)]{Stacks}. Over the regular semisimple open set, $q$ is finite by Lemma~\ref{lem:bundles}, so the evaluation $q^*q_*\OO_Z(-H)\to\OO_Z(-H)$ is surjective there: on affine charts it is the surjection $B\otimes_A N\to N$. The projection formula and $A=q^*\OO(1)$ now identify these sections with sections of $\OO_Z(mA-H)$, proving the assertion. There are finitely many log canonical centers of $(Y,D)$ by Lemma~\ref{lem:pair-facts}(2), and each meets $\pi^{-1}(Z^{\mathrm{rs}})$ by Lemma~\ref{lem:global-boundary}. For each center, sections vanishing identically on its image form a proper linear subspace, by the base-point assertion just proved. Avoiding these finitely many subspaces, choose \[ \Gamma\in|mA-H| \] so that $\pi^*\Gamma$ contains none of those centers. Fix a log resolution of $D+\pi^*\Gamma$, and let $E$ run over its exceptional divisors and the strict transforms of this boundary. Put $a_E=a_E(Y,D)$ and $b_E=\ord_E(\pi^*\Gamma)$. If $a_E=0$, the center of $E$ is a log canonical center, so the choice of $\Gamma$ gives $b_E=0$. For the finitely many remaining $E$ with $b_E>0$, choose a positive rational $t$ with $2tb_E\leq a_E$, and also require $tm