3 Character Sheaves on Disconnected Groups , Iv

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 [L10], [L11], [L12]) which attempts to develop a theory of character sheaves on G. The numbering of the sections and references continues that of the earlier Parts. Assume that k is an algebraic closure of a finite field Fq and that G has a fixed Fq-rational structure with Frobenius map F : G −→ G. To any triple (L, S, E) (where L is a Levi of a parabolic of G, S is an isolated stratum of the normalizer of L, with certain properties, and E is an irreducible cuspidal local system on S) we have an associated in 5.6 a (not necessarily irreducible) intersection cohomology complex K on G. If F (L) = L, F (S) = S and we are given an isomorphism F E −→ E , there is an induced isomorphism φ : F K −→ K hence the characteristic function χK,φ : G F −→ Q̄l is well defined. The main result of this paper (Theorem 21.14) is that the functions χK,φ that are not identically zero (for various (L, S, E) up to G -conjugacy) form a Q̄lbasis of the vector space V of functions G −→ Q̄l that are constant on G conjugacy classes. The proof uses several of the results developed in earlier Parts (the generalized Springer correspondence in §11, the generalized Green functions in §15, the character formula in §16). It also uses the classification of cuspidal local systems (this is needed in §17 which is a preliminary to the proof of Theorem 21.14). A corollary of the main theorem is Theorem 21.21 which states that the characteristic functions of admissible complexes A such that F A ∼= A form a basis for V. In the connected case such a result was proved in [L13] subject to some mild restrictions on the characteristic. The present proof has no restrictions on the characteristic and it makes no use of the orthogonality formulas which will appear in a later stage of the theory. Another corollary of the main theorem is the construction in §22 of a ”twisted induction” map from certain functions on a subgroup of G to functions on G .