4 Character Sheaves on Disconnected Groups ,

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 [L9] which attempts to develop a theory of character sheaves on G. The usual convolution of class functions on a connected reductive group over a finite field makes sense also for complexes in D(G0) and then it preserves (see [Gi]) in the derived sense the class of character sheaves on G. In §32 we define, more generally, a natural convolution operation for parabolic character sheaves (see 32.21(a)). A key role in our study of convolution is played by Theorem 32.6 which describes explicitly the convolution of two basic complexes of the form K̄s,L J,D in terms of multiplication in some Hecke algebra. Using this we define a map which to each parabolic character sheaf associates an orbit of a subgroup of the Weyl group on the set of isomorphism classes of ”tame” local systems of rank 1 on the torus T, see 32.25(b); in fact we define a refinement of this map in 32.25(a). The main result of §33 is Proposition 33.3 (a generalization of [L3, III, 14.2(b)]). It asserts that (under a cleanness assumption), the cohomology sheaves of a character sheaf restricted to an open subset of the support of a different character sheaf are disjoint from the local system given by the second character sheaf on that open subset. (This plays a key role in the argument in 35.22.) In §34 we study the algebra Hn of 31.2 (or rather an extension H D n of it) in the spirit of our earlier study [L12] of a usual Iwahori-Hecke algebra by means of the asymptotic Hecke algebra. This allows us to construct representations of H n starting from representations of H n , the specialization of H D n at v = 1. In 34.19 we define some invariants bA,u of a character sheaf A which depend also on an irreducible representation Eu of H D,1 n . These generalize the invariants cA,E of [L3, III,12.10]. From the definition, bA,u is a rational function in the indeterminate v and one of our goals is to show that bA,u is in fact a constant. This goal is achieved in §35 under a cleanness assumption and a quasi-rationality assumption on Eu. (See Theorem 35.23 which is a generalization of [L3, III, 14.9].) In §35 we prove an orthogonality