UNIPOTENT ALMOST CHARACTERS OF SIMPLE p-ADIC GROUPS

Abstract

0.1. Let G be a simple adjoint algebraic group defined and split over the finite field F[subscript q]. Let K[subscript 0] = [bar over F][subscript q]((ǫ)), K =[bar over F][subscript q]((ǫ)). We are interested in the characters of the standard representations (in the sense of Langlands) of G(K[subscript 0]) corresponding to the (irreducible) unipotent representations ([L6]) of G(K[subscript 0]), restricted to the set G(K[subscript 0])[subscript rsc] = G(K)[subscript rsc] ∩ G(K[subscript 0]) where G(K)[subscript rsc] is the intersection of the set G(K)[subcript rs] of regular semisimple elements in G(K) with the set G(K)[subscript c] of compact elements in G(K) (that is, elements which normalize some Iwahori subgroup of G(K)); we call these restrictions the unipotent characters of G(K[subscript 0]). We hope that the unipotent characters (or some small linear combination of them) have a geometric meaning in the same way as the characters of (irreducible) unipotent representations of G(F[subscript q]) can be expressed in terms of character sheaves on G. Thus we are seeking some geometric objects on G(K)[subscript c] on which the Frobenius map acts and from which the unipotent characters can be recovered.