Character Sheaves and Generalizations

1. Let k be an algebraic closure of a finite field Fq. Let G = GLn(k). The group G(Fq) = GLn(Fq) can be regarded as the fixed point set of the Frobenius map F : G −→ G, (gij) 7→ (g q ij). Let Q̄l 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(Fq) over an algebraically closed field of characteristic 0, which we take to be Q̄l, 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(Fq) can be deduced for any q. This allows us to unify the representation theories of G(Fq) for various q. The geometric objects needed in the theory are provided by intersection cohomology. Let X be an algebraic variety over k, let X0 be a locally closed irreducible, smooth subvariety of X and let E be a local system over X0 (we say ”local system” instead of ”Q̄l-local system”). Deligne, Goresky and MacPherson attach to this datum a canonical object IC(X̄0, E) (intersection cohomology complex) in the derived category D(X) of Q̄l-sheaves on X ; this is a complex of sheaves which extends E to X (by 0 outside the closure X̄0 ofX0) in the most economical possible way so that local Poicaré duality is satisfied. We say that IC(X̄0, E) is irreducible if E is irreducible. Now take X = G and take X0 = Grs to be the set of regular semisimple elements in G. Let T be the group of diagonal matrices in G. For any integer m ≥ 1 invertible in k we have an unramified n!m-fold covering πm : {(g, t, xT ) ∈ Grs × T ×G/T ; x gx = t} −→ Grs, (g, t, xT ) 7→ g. An irreducible local system E on Grs is said to be admissible if it is a direct summand of the local system πm!Q̄l for some m as above. The character sheaves on G are the complexes IC(G, E) for various admissible local systems E on Grs. We show how the irreducible characters of G(Fq) can be recovered from character sheaves on G. If A is a character sheaf on G then its inverse image F A under F is again a character sheaf. There are only finitely many A (up to isomorphism) such that F A is isomorphic to A. For any such A we choose an isomorphism