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 numbering of the sections and references continues that of the earlier Parts. Section 23 is a generalization of results in [L3, II,§7]. It is a preparation for the proof of the orthogonality formulas for certain characteristic functions in Section 24 which generalize those in [L3, II,§9,§10]. Section 25 describes the cohomology sheaves of a class of complexes which includes the admissible complexes on G. In particular we show that these cohomology sheaves restricted to any stratum of G are local systems of a particular kind. In the connected case this reduces to a strengthening of [L3, III,14.2(a)]. In Section 26 we give a variant of the definition of parabolic character sheaves in [L10] in terms of admissibile complexes. Note that even if one is only interested in parabolic character sheaves of connected groups, one cannot avoid using the theory of character sheaves on disconnected groups. In Section 27 we discuss the induction functor. The present treatment differs from one in the connected case, given in [L3, I,§4]. Notation. Let F be a local system on an algebraic variety Y . If k is an algebraic closure of a finite field Fq, F : Y −→ Y is the Frobenius map for an Fq-rational structure and ǫ : F F ∼ −→ F is an isomorphism, we denote by ǫ̌ : F F̌ −→ F̌ the unique isomorphism such that for any y ∈ Y , ǫ̌ : ĚF (y) −→ Ěy is the isomorphism transpose inverse to ǫ : EF (y) −→ Ey. If X is an algebraic variety and A ∈ D(X) is IC(X ,F) extended by 0 on X−X ′ where X ′ is a closed irreducible subvariety of X and F is a local system on an open dense smooth subvariety X ′ 0 of X , let Ǎ = IC(X , F̌) extended by 0 on X−X . We have also Ǎ = D(A)[−2 dimX ]. If k is an algebraic closure of a finite field Fq , F : X −→ X is the Frobenius map for an Fq-rational structure such that F (X ) = X , F (X ′ 0) = X ′ 0 and α : F A ∼ −→ A is an isomorphism which restricts to ǫ : F F ∼ −→ F over X0, we denote by α̌ : F Ǎ −→ Ǎ the unique isomorphism which restricts to ǫ̌ : F F̌ ∼ −→ F̌ over X0.