\documentclass[11pt]{amsart} \usepackage{amsmath,amssymb,mathrsfs,indentfirst} \usepackage[pdfusetitle,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} \urlstyle{same} \renewcommand{\keywordsname}{Keywords} \newtheorem{maintheorem}{Theorem} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \setlength{\parindent}{15pt} \setlength{\emergencystretch}{1em} \allowdisplaybreaks[2] \raggedbottom \setcounter{tocdepth}{1} \title[Lusztig's conjectures in types B and C]{Lusztig's conjectures for \(B_n\), \(\widetilde B_n\), and \(\widetilde C_n\)} \author{Qiyue Tang} \date{November 8, 2026} \keywords{Unequal-parameter Hecke algebras, Kazhdan--Lusztig cells, asymptotic algebras, Plancherel measure} \subjclass[2020]{Primary 20C08. Secondary 20F55} \begin{document} \begin{abstract} We prove Lusztig's conjectures \({\rm P1}\)--\({\rm P15}\) for every positive integer-valued weight function on the finite Coxeter group of type \(B_n\) and the nonextended affine Weyl groups of types \(\widetilde B_n\) and \(\widetilde C_n\). \end{abstract} \maketitle \tableofcontents \section{Introduction} Kazhdan and Lusztig introduced the canonical basis of the equal-parameter Hecke algebra in their study of Weyl group representations \cite{KazhdanLusztig}. The coefficients of multiplication in this basis define left, right, and two-sided preorders on the Coxeter group. Passing to an equivalence class for the left preorder gives a subquotient of the left regular module. In type \(A\), the Robinson--Schensted correspondence describes the cells, linking these subquotients to the representation theory of symmetric groups. In his work on left cells in Weyl groups, Lusztig allowed different parameters on the simple reflections and studied the resulting representations in connection with primitive ideals \cite{LusztigLeft}. In the study of affine Weyl groups, Lusztig introduced the \(a\)-function, which records the highest exponent at which a fixed basis element can occur in a product \cite{LusztigAffine}. The coefficients at those exponents give the multiplication of the asymptotic algebra. Their relation to the Kazhdan--Lusztig preorders governs the decomposition of this algebra by two-sided cells. In the equal-parameter theory, Shi described the cells in affine type \(A\) \cite{ShiAffine}, and Lusztig identified the two-sided cells of affine Weyl groups with unipotent conjugacy classes in the associated complex groups \cite[Theorem~4.8]{LusztigAffineIV}. Lusztig organized the required degree bounds, symmetries, and compatibility statements as properties \({\rm P1}\)--\({\rm P15}\) in his monograph \cite[\S14.2]{Lusztig}. Among them, P4 orders the \(a\)-values along the two-sided preorder, P12 compares the \(a\)-function with that of a parabolic subgroup, and P15 compares left and right multiplication with independent parameters on a fixed degree layer. In the split case, positivity and boundedness give the properties by \cite[\S15]{Lusztig}. The Hodge theory of Soergel bimodules gives the equal-parameter positivity theorem for Coxeter systems \cite{EliasWilliamson}. The quasisplit case in \cite[\S16]{Lusztig} obtains unequal weights from the restriction of the length function to a fixed-point Coxeter group. For general positive weights the multiplication coefficients can have both signs. In particular, arguments using positivity to rule out cancellation require a different justification. Chen and Hu proved boundedness for all positively weighted Coxeter systems with finitely many simple reflections \cite[Theorem~1.3]{ChenHu}. This gives a uniform upper bound on the degrees. For finite type \(B_n\), Bonnaf\'e and Iancu described left cells in the asymptotic regime by a signed Robinson--Schensted correspondence \cite{BonnafeIancu}. Geck and Iancu computed the \(a\)-function in that regime \cite{GeckIancu}, and Geck proved the fifteen properties in the generic asymptotic case \cite[Theorem~5.3]{GeckFinite}. For general unequal weights, Bonnaf\'e, Geck, Iancu, and Lam related cells to domino insertion. They showed that, away from the ratios \(1,\ldots,n-1\) of the short weight to the long weight, the leading matrix coefficients lie in \(\{0,1,-1\}\), and the indices supporting each leading column lie in a single left cell \cite[Theorem~2.1 and Proposition~2.3]{BGIL}. They conjectured that the left cells in these parameter ranges are the recording fibres of domino insertion \cite[Conjecture~A]{BGIL}. The complete result for finite \(B_3\) appears in Bonnaf\'e's treatment of rank three and in Gao and Xie's theorem for Coxeter systems with three simple reflections \cite[\S25.1A]{BonnafeBook} and \cite[Theorem~9.1]{GaoXie}. Unequal weights also arise in Lusztig's construction of weighted affine Weyl groups attached to unipotent blocks \cite[\S\S5, 7]{LusztigBlocks}. For affine Weyl groups with unequal weights, Xie established the properties needed for the based ring of the lowest two-sided cell and determined that ring \cite{XieLowest}. Guilhot and Parkinson proved all fifteen properties for all positive weights in types \(\widetilde G_2\) and \(\widetilde C_2\), using balanced representations and the affine Hecke Plancherel formula \cite{GuilhotParkinsonG,GuilhotParkinsonC}. Dawydiak used the Plancherel formula to bound the denominators of the coefficients of the inverse asymptotic map in the equal-parameter case \cite{Dawydiak}. The formula has also been used to determine the asymptotic algebra in equal-parameter extended affine type \(A\) \cite{CLGLP}. For equal-parameter extended type \(\widetilde B_n\), Parkinson gave a factorization of elements and computed the asymptotic algebra of the second lowest two-sided cell \cite{ParkinsonSecond}. \begin{maintheorem}\label{thm:main} Lusztig's properties \({\rm P1}\)--\({\rm P15}\) hold for every positive integer-valued weight function on the finite Coxeter system \(B_n\) and the nonextended affine Weyl systems \(\widetilde B_n\) and \(\widetilde C_n\), in the following ranks: \[ B_n\quad(n\geq 1),\qquad \widetilde B_n\quad(n\geq 3),\qquad \widetilde C_n\quad(n\geq 2). \] \end{maintheorem} The Coxeter system \(\widetilde B_2\) is isomorphic to \(\widetilde C_2\). In rank one the affine system, also denoted \(\widetilde C_1\), is infinite dihedral. Its unequal-parameter case is \cite[\S17]{Lusztig}, and its equal-parameter case is split. In finite type \(B_n\) the proof uses an induction that carries the off-wall cell lattices together with the fifteen properties. A comparison with standard modules for the rational Cherednik algebra gives a filtration of the original regular module. At integral parameter ratios the filtration is obtained by specialization and an integral projection argument. In affine type the Plancherel formula assigns a degree to each tempered inducing family. We use finite parabolic subgroups to find a constituent of that degree in every irreducible component of the family. Integral projections onto spectral levels and relative canonical induction then identify the spectral filtration with ideals spanned by the original Kazhdan--Lusztig elements. An integral trace inequality in the resulting leading algebra gives the distinguished elements. For \(\widetilde B_n\) the finite-parabolic witnesses and the trace inequality come from an auxiliary \(\widetilde C_n\) algebra with one endpoint weight zero. The filtration is constructed in the \(\widetilde B_n\) algebra itself. \section{Hecke algebras and the fifteen properties}\label{sec:prelim} Let \((W,S)\) be a Coxeter system with length function \(\ell\) and Bruhat order \(<\). A weight function \(L:W\to\mathbb Z\) satisfies \(L(xy)=L(x)+L(y)\) whenever \(\ell(xy)=\ell(x)+\ell(y)\). We take \(L(s)>0\) for \(s\in S\), except in the auxiliary \(\widetilde C_n\) algebra with affine endpoint weight zero. Put \(\mathcal A=\mathbb Z[v,v^{-1}]\). The Hecke algebra \(H\) has normalized standard basis \((N_w)_{w\in W}\), with length-additive multiplication and quadratic relations \[ (N_s-v^{L(s)})(N_s+v^{-L(s)})=0. \] The bar involution is the semilinear algebra involution sending \(v\) to \(v^{-1}\) and \(N_w\) to \(N_{w^{-1}}^{-1}\). The positive Kazhdan--Lusztig basis consists of its fixed elements \[ C_w=N_w+\sum_{y1\) we extend \(*\) conjugate-linearly. The form \(\langle h,k\rangle=\tau(hk^*)\) then has orthonormal basis \((N_w)\). Write \(h_{x,y,z}\) for the coefficient of \(C_z\) in \(C_xC_y\). The left preorder \(\leq_L\) is the reflexive transitive closure of the arrows from \(y\) to \(z\) for which \(h_{s,y,z}\ne 0\) for some \(s\in S\), with \(z\leq_L y\) for such an arrow. The right preorder \(\leq_R\) is defined by right multiplication. The two-sided preorder \(\leq_{LR}\) is generated by both. The equivalence classes for the symmetric parts of these preorders are the left, right, and two-sided cells, respectively. For a Laurent polynomial \(f\), write \([v^j]f\) for its coefficient at the integer exponent \(j\) and \(\deg_v f\) for its largest exponent, with \(\deg_v0=-\infty\). Set \[ a(z)=\sup_{x,y\in W}\deg_v h_{x,y,z}. \] Whenever this number is finite the supremum is a maximum, since the degrees are integral and \(C_1 C_z=C_z\). For strictly positive weights, \(\tau(C_z)\ne0\) by \cite[Proposition~5.4(a)]{Lusztig}. In this case write \(\Delta(z)=-\deg_v\tau(C_z)\), \(n_z=[v^{-\Delta(z)}]\tau(C_z)\), and \[ \mathcal D=\{z\in W:a(z)=\Delta(z)\}, \qquad \gamma_{x,y,z}=[v^{a(z^{-1})}]h_{x,y,z^{-1}}. \] Inversion preserves \(a\) by the anti-involution. The auxiliary algebra with a zero endpoint is used only for restrictions and trace calculations. The statements to prove, with the labels of \cite[\S14.2]{Lusztig}, are as follows. \begin{enumerate} \item[\({\rm P1}\)] \(a(z)\leq \Delta(z)\). \item[\({\rm P2}\)] If \(d\in\mathcal D\) and \(\gamma_{x,y,d}\ne0\), then \(x=y^{-1}\). \item[\({\rm P3}\)] For every \(y\) there is a unique \(d\in\mathcal D\) with \(\gamma_{y^{-1},y,d}\ne0\). \item[\({\rm P4}\)] If \(z'\leq_{LR}z\), then \(a(z')\geq a(z)\). \item[\({\rm P5}\)] The nonzero coefficient in P3 is \(n_d=\pm1\). \item[\({\rm P6}\)] For every \(d\in\mathcal D\), \(d^2=1\). \item[\({\rm P7}\)] \(\gamma_{x,y,z}=\gamma_{y,z,x}\). \item[\({\rm P8}\)] If \(\gamma_{x,y,z}\ne0\), then \(x\sim_L y^{-1}\), \(y\sim_L z^{-1}\), and \(z\sim_L x^{-1}\). \item[\({\rm P9}\)] If \(z'\leq_L z\) and \(a(z')=a(z)\), then \(z'\sim_L z\). \item[\({\rm P10}\)] The right-hand version of P9 holds. \item[\({\rm P11}\)] The two-sided version of P9 holds. \item[\({\rm P12}\)] For \(y\) in a standard parabolic subgroup \(W_I=\langle I\rangle\), \(I\subseteq S\), its intrinsic \(a\)-value is its ambient \(a\)-value. \item[\({\rm P13}\)] Each left cell contains a unique \(d\in\mathcal D\), and \(\gamma_{y^{-1},y,d}\ne0\) for every \(y\) in that cell. \item[\({\rm P14}\)] \(z\sim_{LR}z^{-1}\) for every \(z\). \item[\({\rm P15}\)] If \(a(w)=a(y)\), then for any \(x,x'\in W\), \[ \sum_{z\in W}h_{w,x',z}(u)h_{x,z,y}(v) =\sum_{z\in W}h_{x,w,z}(v)h_{z,x',y}(u) \] in \(\mathbb Z[u,u^{-1},v,v^{-1}]\). Here \(h(u)\) denotes the same Laurent polynomial with indeterminate \(u\). Both sums have finite support because each product of basis elements has finite support. \end{enumerate} When P4 holds, the spans of the \(C_z\) with \(a(z)\geq A\) and with \(a(z)>A\) are two-sided ideals for each integer \(A\). Indeed multiplication by \(C_s\), for \(s\in S\), respects the two thresholds by the definition of the preorders, and the \(C_s\) generate \(H\). On the quotient of the former ideal by the latter, apply the left action with indeterminate \(v\) and the right action with indeterminate \(u\) to the common coordinate basis indexed by \(a(z)=A\). The equality in P15 says that these actions commute. In either sum a nonzero term between endpoints of value \(A\) has intermediate index of value \(A\) by P4. In type \(B_n\) the weight on the adjacent transpositions is \(a>0\) and the short endpoint has weight \(b>0\). In affine type \(\widetilde C_n\) the middle weights are \(a\), the finite endpoint has weight \(b\), and the affine endpoint has weight \(c\). In type \(\widetilde B_n\) the fork and the chain up to the short endpoint have weight \(a\), and the short endpoint has weight \(b\). The notation \(a(w)\) always denotes the function defined by multiplication, whereas \(a\) without an argument denotes the common middle weight. \input{double-induction} \input{finite-b} \input{spectral} \input{facet-attainment} \input{affine-b} \input{affine-kl} \input{distinguished} \input{conclusion} \input{references} \begingroup\small \noindent\textsc{Zhili College, Tsinghua University, Beijing 100084, China}\par \noindent\textit{Email address:}\enspace \mbox{\href{mailto:tangqy24@mails.tsinghua.edu.cn}{tangqy24@mails.tsinghua.edu.cn}}\par \endgroup \end{document}