Character Sheaves on Disconnected Groups

Throughout this paper, G denotes a fixed, not necessarily connected, reductive algebraic group over an algebraically closed field k with a fixed connected component D which generates G. This paper is a part of a series [L9] which attempts to develop a theory of character sheaves on D. Our main result here is the classification of ”unipotent” character sheaves on D (under a mild assumption on the characteristic of k). This extends the results of [L3, IV,V] which applied to the case where G = G. While in the case of G = G the classification of unipotent character sheaves is essentially the same as the classification of unipotent representations of a split connected reductive group over Fq, the classification in the general case is essentially the same as the classification of unipotent representations of a not necessarily split connected reductive group over Fq given in [L14]. We now describe the content of the various sections in more detail. §43 contains some preparatory material concerning (extended) Hecke algebra and twosided cells which are used later in the study of unipotent character sheaves. In §44 we study the unipotent character sheaves in connection with Weyl group representations and two-sided cells. (But it turns out that the method of associating a two-sided cell to a unipotent character sheaf along the lines of [L3, III] is better for our purposes than the one in §41.) A number of results in this section are conditional (they depend on a cleanness property and/or on a parity property); they will become unconditional in §46. In §45 we show that the problem of classifying the unipotent character sheaves on D can be reduced to the analogous problem in the case where G is simple and G has trivial centre. In §46 we extend the results of [L3, IV,V] on the classification of unipotent character sheaves on D from the case G = G to the general case. Erratum to [L9, V]; in line 4 of 25.1 replace last a by s. Erratum to [L9, VI]: on p.383 l.-25,-24 replace Z by Z̄ and ∆j by ∆j . Erratum to [L9, VII]: on p.248, l.4 of 35.5 replace GF by G .