A ug 2 00 3 CHARACTER SHEAVES ON DISCONNECTED GROUPS , III

Throughout this paper, G denotes a fixed, not necessarily connected, reductive algebraic group over an algebraically closed field k. This paper is a part of a series (beginning with [L9],[L10]) which attempts to develop a theory of character sheaves on G. Assume now that k is an algebraic closure of a finite field Fq and that G has a fixed Fq-structure with Frobenius map F . Let (L, S, E , φ0) be a quadruple where L is an F -stable Levi of some parabolic of G, E is a local system on an isolated F -stable stratum S of NGL with certain properties and φ0 is an isomorphism of E with its inverse image under the Frobenius map. To (L, S, E) we have associated in 5.6 an intersection cohomology complex K = IC(Ȳ , π!Ẽ) on G. Moreover, φ0 gives rise to an isomorphism φ between K and its inverse image under the Frobenius map. There is an associated characteristic function χK,φ (see 15.12(a)) which is a function G −→ Q̄l, constant on (G ) -conjugacy classes. The main result of this paper is Theorem 16.14 which shows that the computation of this function can be reduced to an analogous computation involving only unipotent elements in a smaller group (the centralizer of a semisimple element). (This is a generalization of [L8, Theorem 8.5]. However, even if G is assumed to be connected, as in [L8], our Theorem 16.14 is more general than that in [L8], since here we do not make the assumption that E is cuspidal. Also, unlike the proof in [L8], the present proof does not rely on the classification of cuspidal local systems.) A main ingredient in Theorem 16.14 are the generalized Green functions, see 15.12(c), which generalize those in [L8, 8.3.1]. One of the key properties of the generalized Green functions is the invariance property 15.12(d). In the connected case, this property was stated in [L8, 8.3.2], but the proof given there was incomplete (as pointed out by F.Letellier). Most of Section 15 is devoted to establishing this invariance property.