Green functions and character sheaves
type: Item
instance of: scholarly article
title: Green functions and character sheaves
zbMATH Open document ID: 0695.20024
DOI: 10.2307/1971496
author: G. Lusztig
published in: Annals of Mathematics. Second Series
publication date: 1990-01-01T00:00:00Z
review text: Let G be a connected, reductive algebraic group defined over \(F_ q\), and let \(F: G\to G\) be a Frobenius morphism. Let \(G^ F\) be the finite group of F-fixed points of G. Corresponding to each F-stable maximal torus T, \textit{P. Deligne} and the author [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] defined functions \(Q^ G_ T\), known as Green functions, on the set of unipotent elements of \(G^ F\). Later the author gave a new definition of Green functions using intersection cohomology and in his work on character sheaves [Adv. Math. 61, 103-155 (1986; Zbl 0602.20036)] gave a method of computing these functions in principle. The coincidence of the two definitions of Green functions was known for large p (where q is a power of p) by work of Springer and Kazhdan, and for all ``good'' p in some cases by further work of the author [J. Algebra 104, 146-194 (1986; Zbl 0603.20037)]. In this paper the author proves that the two definitions coincide for all p, provided q is sufficiently large. More generally, he considers the generalized Green functions defined by him via intersection cohomology [in Adv. Math. 57, 226-265 (1985; Zbl 0586.20019)] and connects them with functions obtained via twisted induction from a Levi subgroup of G, when p is ``almost good''. Let \({\mathcal D}G\) denote the bounded derived category of constructible \(\bar Q_{\ell}\)-sheaves on G (here \(\ell\) is a prime distinct from p). The abelian category \({\mathcal M}G\) of perverse sheaves on G is a full subcategory of \({\mathcal D}G\), and character sheaves are certain objects in \({\mathcal M}G\). In analogy with the Harish-Chandra theory for \(G^ F\), certain character sheaves are called cuspidal. If K is a complex in \({\mathcal D}G\) with a given isomorphism \(\phi\) : \(F^*K\overset \sim \rightarrow K\), then one gets a \(\bar Q_{\ell}\)-valued characteristic function \(\chi_{K,G}\) on \(G^ F\) by setting \(\chi_{K,\phi}(x)=\sum_{i\geq 0}(-1)^ iTr(\phi,H^ i_ xK)\) where \(H^…
reviewed by: Bhama Srinivasan
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14L40
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4138040
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: Frobenius morphism
zbMATH Keywords: unipotent elements
zbMATH Keywords: Green functions
zbMATH Keywords: intersection cohomology
zbMATH Keywords: character sheaves
zbMATH Keywords: twisted induction
zbMATH Keywords: Levi subgroup
zbMATH Keywords: derived category of constructible \(\bar Q_{\ell}\)-sheaves
zbMATH Keywords: perverse sheaves
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971496
OpenAlex ID: W2010761433
description: scientific article; zbMATH DE number 4138040