Green functions and a conjecture of Geck and Malle
type: Item
instance of: scholarly article
title: Green functions and a conjecture of Geck and Malle
zbMATH Open document ID: 0962.20036
author: Toshiaki Shoji
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-05-02T00:00:00Z
full work available at URL: https://eudml.org/doc/120973
full work available at URL: BAG
review text: Let \(W\) be a finite Coxeter group. \textit{G.~Lusztig} has attached to each irreducible representation \(E\) of \(W\) a polynomial \(D_E\in\mathbb{R}[X]\) (where \(X\) is an indeterminate) called the formal or generic degree of \(E\) and two functions: the \(a\)-function is defined as \(E\mapsto a_E\), where \(a_E\) is the greatest integer \(n\geq 0\) such that \(X^n\) divides the polynomial \(D_E\), the \(b\)-function \(b(E)\) is defined as \(E\mapsto b_E\) where \(b_E\) is the smallest integer \(m\geq 0\) such that \(E\) occurs with non-zero multiplicity in \(V^{\otimes m}\), where \(V\) is an \(\mathbb{R} W\)-module affording the reflection character of \(W\) (see [Characters of reductive groups over a finite field, Ann. Math. Stud. 107 (1984; Zbl 0556.20033)]). We always have \(a_E\leq b_E\), and \(E\) is called special if the equality holds. The set \(\text{Irr}(W)\) of isomorphism classes of irreducible representations of \(W\) is partioned into various two-sided cells of \(W\). Each two-sided cell contains a unique special representation of \(W\). Let \(G\) be a connected reductive group defined over an algebraic closure \(k\) of a finite field \(\mathbb{F}_q\) with \(q\) elements. Assume that \(W\) is the Weyl group of \(G\). The Springer correspondence defines a natural injective map from \(\text{Irr}(W)\) to the set of the pairs \((C,{\mathcal L})\), where \(C\) is a unipotent class in \(G\) and \(\mathcal L\) is a \(G\)-equivariant irreducible \(\overline\mathbb{Q}_\ell\)-local system on \(C\). A unipotent class \(C\) is called special if \((C,\overline\mathbb{Q}_\ell)\) corresponds to a special representation of \(W\). For any special unipotent class \(C\) in \(G\), the \(\mathcal L\) special piece \(\widehat C\) is defined as the subset of the unipotent variety \(G_{\text{uni}}\) consisting of all the elements in the closure \(\overline C\) of \(C\) which are not in the closure of any special unipotent class \(C'\) such that \(C'\subset\overline C\) a…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 1428666
zbMATH Keywords: finite reductive groups
zbMATH Keywords: unipotent classes
zbMATH Keywords: Springer correspondence
zbMATH Keywords: Green functions
zbMATH Keywords: Coxeter groups
zbMATH Keywords: irreducible representations
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1428666