Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank <8. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams of the nilpotent orbits in the Langlands dual group.
For a $p$-adic field $F$, the embeddings of a tame supercuspidal representation of $G= {\rm GL}_n (F)$ in the space of smooth functions on the set of symmetric matrices in $G$ are determined.
Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished c…
Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relati…
Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $σ$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $π$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $π^σ\simeqπ^{\vee}$ (such a representation is said to be $σ$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times…
Let $\mathrm{G} = \mathrm{Gl}_{n}(K)$, and $\mathrm{H} = \mathrm{G}^σ$ for $σ$ an involution of the form $g\rightarrow aga^{-1}$, It is known that for $K =\mathbb{Q}_q$ any irreducible representation of $\mathrm{G}$ with an $\mathrm{H}$ invariant functional, is self dual, we prove an analogous result for $\mathbb{F}_{q}$.
Let $E/F$ be a quadratic extension of fields, and $G$ a connected quasi-split reductive group over $F$. Let $G^{op}$ be the opposition group obtained by twisting $G$ by the duality involution considered by Prasad. Assume that the field $F$ is finite. Let $π$ be an irreducible generic representation of $G(E)$. When $π$ is a Shintani base change lift of some representation of $G^{op}(F)$, we give an explicit nonzero $G(F)$-invariant vector in terms of the Whittaker vector of $π$. This shows particularly that $π$ is $G(F)$-distinguished. When the field $F$ is $p$-adic, the paper also proves that …
This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal repr…
Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.
Let $(W,S)$ be any Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have shown that the corresponding set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as a "twisted" analogue of the much-studied family of Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coeffi…
Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $σ$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove the following result: an irreducible representation $π$ of $U$ is $U^σ$-distinguished if and only if it is $σ$-self-dual and in this case $\mathrm{Hom}_{U^σ}(π,\mathbb{C})$ has dimension one. When $σ$ is a Galois involution these results imply a bijective correspondence between the set $\mathrm{Irr}(U^σ)$ of isomorphism classes of irreducible representations of …
Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^σ_{y,w}$ which one can view as "twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^σ_{y,w}$ can have negative coefficients, but displa…
Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced degree d such that rd=2n. Let K be a quadratic extension of F in the algebra of matrices of size r with coefficients in D, and H be its centralizer in G. We study selfdual cuspidal representations of G and their distinction by H from the point of view of type theory. Given such a representation pi of G, we compute the value at 1/2 of the epsilon factor of the…
The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set $\D$ has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of $\D$ we assign an explicitly defined set of equivalence relations on $W$ that altogether conjecturally…
The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.
We introduce the computer algebra package {\sf PyCox}, written entirely in the {\sf Python} language. It implements a set of algorithms - in a spirit similar to the older {\sf CHEVIE} system - for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan--Lusztig cells and $W$-graphs, which works efficiently for all finite groups of rank $\leq 8$ (except $E_8$). We also discuss the computation of Lusztig's leading coefficients of character values and distinguished involutions (which works for $E_8$ as well). Our experiments…
Affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an affine generalization of Robinson-Schensted correspondence. We show that AMBC gives a simple way to compute a distinguished (or Duflo) involution in each Kazhdan-Lusztig cell of affine symmetric groups. We then use AMBC to give the first known canonical presentation for the asymptotic Hecke algebras of extended affine symmetric groups. As an application, we show that AMBC gives a conceptual way to compute Lusztig-Vogan bijection. For the latter we build upon prior works of Achar a…
