Character Sheaves and Generalizations

Abstract

Let k be an algebraic closure of a finite field F _q. Let G = GL _n(k). The group G(F _q) = GL _n(F _q) can be regarded as the fixed point set of the Frobenius map F: G → G,$(g_{ij} ) \mapsto (g_{ij}^q )$. Let $\mathop {\mathbf{Q}}\limits^{\_\_} _l$ be an algebraic closure of the field of l-adic numbers, where l is a prime number invertible in k. The characters of irreducible representations of G(F _q) over an algebraically closed field of characteristic 0, which we take to be $\mathop {\mathbf{Q}}\limits^{\_\_} _l$, have been determined explicitly by J. A. Green [G]. The theory of character sheaves [L2] tries to produce some geometric objects over G from which the irreducible characters of G(F _q) can be deduced for any q. This allows us to unify the representation theories of G(F _q) for various q. The geometric objects needed in the theory are provided by intersection cohomology.