Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

AssumeG is a connected reductive algebraic group defined over an algebraic closure K = Fp of the finite field of prime order p > 0. Furthermore, assume that F : G → G is a Frobenius endomorphism of G. In this article we give a formula for the value of any F -stable character sheaf of G at a unipotent element. This formula is expressed in terms of class functions of GF which are supported on a single unipotent class of G. In general these functions are not determined, however, we give an expression for these functions under the assumption that Z(G) is connected, G/Z(G) is simple and p is a good prime for G. In this case our formula is completely explicit.