On the character values of finite Chevalley groups at unipotent elements
type: Item
instance of: scholarly article
title: On the character values of finite Chevalley groups at unipotent elements
zbMATH Open document ID: 0603.20037
DOI: 10.1016/0021-8693(86)90245-0
author: G. Lusztig
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(G(F_ q)\) be the group of rational points of a reductive connected algebraic group G over a finite field \(F_ q\). The author presents a method for computing the values of the irreducible characters of \(G(F_ q)\) at unipotent elements. The author's method can be explained as follows. The space of class functions on \(G(F_ q)\) has two natural orthonormal bases: one, \((f_ i)\), consists of the irreducible characters, the other one, \((f_ j')\) consists of the characteristic functions of certain irreducible perverse sheaves on G, called character sheaves. Then to compute the \((f_ i)\) on the unipotent elements it is enough to solve two other problems: (a) compute the \(f_ j'\) at unipotent elements. (b) compute the transition matrix between the bases \((f_ i)\), \((f_ j').\) Problem (a) is closely related to the problem of computing the local intersection cohomology of the closure of a unipotent class of G with coefficients in a G-equivariant irreducible local system. Problem (b) is more difficult. There is a known formula which expresses those \(f_ i\) which are components of a principal series representation in terms of the \(f_ j'\). This information is quite powerful for solving problem (b). The author asserts that his method can be used to determine the Green functions of \(G(F_ q)\) assuming that G is adjoint simple, split over \(F_ q\) and q is subject to the following conditions: (*) Type A: no condition; type B: q odd; type C, \(D: q\equiv 1 (mod 4)\); type \(G_ 2\), \(E_ 6:\) \(q\equiv 1 (mod 6)\); type \(F_ 4\), \(E_ 7:\) \(q\equiv 1 (mod 12)\); type \(E_ 8:\) \(q\equiv 1 (mod 60).\) The author also asserts that he can use his method to compute for G as above, the character at unipotent elements of any irreducible representation of \(G(F_ q)\) such that the corresponding semisimple element in the dual group \(G^*(F_ q)\) has centralizer of maximal semisimple rank. These include, for example, all unipotent representations of \(G(F_ q)\). The author …
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 3973161
zbMATH Keywords: group of rational points
zbMATH Keywords: reductive connected algebraic group
zbMATH Keywords: irreducible characters
zbMATH Keywords: unipotent elements
zbMATH Keywords: class functions
zbMATH Keywords: orthonormal bases
zbMATH Keywords: characteristic functions
zbMATH Keywords: irreducible perverse sheaves
zbMATH Keywords: character sheaves
zbMATH Keywords: local intersection cohomology
zbMATH Keywords: principal series representation
zbMATH Keywords: Green functions
zbMATH Keywords: unipotent representations
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90245-0
OpenAlex ID: W1979583426
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description: scientific article; zbMATH DE number 3973161