Character sheaves. I
type: Item
instance of: scholarly article
title: Character sheaves. I
zbMATH Open document ID: 0586.20018
DOI: 10.1016/0001-8708(85)90034-9
author: G. Lusztig
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The author constructs a geometric theory of characters of a reductive algebraic group G defined over an algebraically closed field. This theory is as close as possible to the theory of irreducible (complex) characters of the corresponding group \(G(F_ q)\) over a finite field \(F_ q\) and yet it should have a meaning over an algebraically closed field. The basic objects of the theory are certain irreducible (l-adic) perverse sheaves [cf. \textit{A. A. Beilinson}, \textit{J. Bernstein}, and \textit{P. Deligne}, Faisceaux pervers (Astérisque 100, 1982; Zbl 0536.14011)] on G; they are the analogues of the irreducible (l-adic) representations of \(G(F_ q)\) and are called the character sheaves of G. The definition of character sheaves is suggested by the following result: any irreducible representation of \(G(F_ q)\) appears in at least one of the virtual representations \(R^{\theta}(w)\), defined by \textit{P. Deligne} and \textit{G. Lusztig} [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. The main object of the paper is to classify the character sheaves of G and to compute their cohomology sheaves. The paper is organized as follows: Section 1 collects some of the basic results on perverse sheaves due to Beilinson-Bernstein-Deligne-Gabber. Section 2 contains the definition of character sheaves. In sections 3 and 4 the author studies the restriction and induction for character sheaves. Section 5 contains some technical preliminaries to section 6. The most difficult result of the paper is theorem 6.9(a) which asserts that the restriction functor carries a character sheaf to a direct sum of character sheaves.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14L40
zbMATH DE Number: 3939580
zbMATH Keywords: reductive algebraic group
zbMATH Keywords: perverse sheaves
zbMATH Keywords: character sheaves
zbMATH Keywords: irreducible representation
zbMATH Keywords: virtual representations
zbMATH Keywords: cohomology sheaves
zbMATH Keywords: restriction
zbMATH Keywords: induction
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
full work available at URL: 0001-8708(85)90034-9
OpenAlex ID: W4210362956
Wikidata QID: Q29393602
cites work: Q3320507
cites work: La conjecture de Weil. II
cites work: Representations of reductive groups over finite fields
cites work: Intersection cohomology complexes on a reductive group
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description: scientific article; zbMATH DE number 3939580