Character sheaves. I
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.
scientific article; zbMATH DE number 3939580