20 06 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. One of the main constructions in [L3] (going back to [L14]) was a procedure which to any character sheaf on G associates a certain two-sided cell in an (extended) Coxeter group. A variant of this construction (restricted to ”unipotent” character sheaves) was later given by Grojnowski [Gr]. Here we give a construction which generalizes that in [L3] (and takes into account the approach in [Gr]) which to any (parabolic) character sheaf on ZJ,D associates a certain type of two-sided cell. The paper is organized as follows. In Section 40 we study certain equivariant sheaves on G/U × G/U (where U is the unipotent radical of a Borel in G) under the convolution operation. Some results in this section are implicit in [L14, Ch.1]. In Section 41 we study the character sheaves on Z∅,D (where D is a connected component of G) by connecting them with sheaves on G/U×G/U. We use this study to attach a two-sided cell to any character sheaf on ZJ,D. (See 41.4.) In Section 42 we study the interaction between the duality operation d (see 38.10, 38.11) and the functor f∅,I (see 36.4). The main result in this section is Proposition 42.9 which contains [L3, III, Cor.15.8(b)] as a special case (with G = G, v = 1). Notation We fix a 1-dimensional Q̄l-vector space V with a given isomorphism V ⊗2 ∼ −→ Q̄l(1) (Tate twist of Q̄l). For n ∈ N we set Q̄l(n/2) = V . For n ∈ Z, n < 0 let Q̄l(n/2) be the dual space of Q̄l(−n/2). If X is an algebraic variety and A ∈ D(X), n ∈ Z we write A[[n/2]] instead of A[n](n/2). (When n is even this agrees with the notation in [L9, II, p.73].)