Character sheaves on nonconnected groups
type: Item
instance of: scholarly article
title: Character sheaves on nonconnected groups
zbMATH Open document ID: 0870.20027
author: Yanyan Li
published in: Journal of Algebra
publication date: 1997-07-08T00:00:00Z
review text: The author studies character sheaves for nonconnected reductive algebraic groups defined over an algebraic closure \(\overline\mathbb{F}_q\) of a finite field \(\mathbb{F}_q\). Let \(G\) be a connected reductive algebraic group defined over \(\mathbb{F}_q\). We let \(F\colon G\to G\) be the corresponding Frobenius map and \(\sigma\colon G @>\sim>>G\) be a semisimple automorphism, of finite order and commuting with \(F\). The author considers the nonconnected group \(G\rtimes\langle\sigma\rangle\) (or more precisely its connected component \(G\cdot\sigma\)) and proves a character formula for the characteristic functions of perverse sheaves on \(G\cdot\sigma\) which are induced from \(T\)-equivariant tame local systems on \(T\cdot\sigma\) (here \(T\) is a maximal torus of \(G\) which is \(\sigma\)-stable and contained in a \(\sigma\)-stable Borel subgroup of \(G\)). By comparison with the character formula for the generalized Deligne-Lusztig induction functor \(R_{T\cdot\sigma}^{G\cdot\sigma}\) he gets an equality which relates the characteristic functions of induced perverse sheaves to generalized Deligne-Lusztig induction (that equality is analogous to those obtained by Lusztig in case of connected groups). He then shows a partial analog for nonconnected groups of Lusztig's conjecture (that conjecture has been proved by Shoji), which says that the characteristic functions of character sheaves are in an essential way the same as the ``almost characters'' defined by Lusztig. One purpose of the paper under review is also to prove a formula which makes precise the effect of Shintani descent \(\text{Sh}_{F,\sigma^{-1}F}\) on the characteristic functions of \(G\)-equivariant \(F\)-stable perverse sheaves on \(G\cdot\sigma\) (in this part of the paper the automorphism \(\sigma\) is not assumed to be semisimple). Finally the author shows that the functors \(R_{T\cdot\sigma}^{G\cdot\sigma}\) and \(\text{Sh}_{F,\sigma^{-1}F}\) commute.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 930154
zbMATH Keywords: character sheaves
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: Frobenius maps
zbMATH Keywords: semisimple automorphisms
zbMATH Keywords: character formula
zbMATH Keywords: characteristic functions
zbMATH Keywords: generalized Deligne-Lusztig induction
zbMATH Keywords: perverse sheaves
zbMATH Keywords: Shintani descent
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: jabr.1996.0272
OpenAlex ID: W2001745378
DOI: 10.1006/JABR.1996.0272
description: scientific article; zbMATH DE number 930154