Character Sheaves , Ii 3

Let G be a connected reductive algebraic group over an algebraically closed field k. Let Z be the algebraic variety consisting of all triples (P, P , UP ′gUP ) where P, P ′ run through some fixed conjugacy classes of parabolics in G and g is an element of G that conjugates P to a parabolic in a fixed ”good” relative position y with P ′ (here UP , UP ′ are the unipotent radicals of P, P ). The varieties Z include more or less as a special case the boundary pieces of the De ConciniProcesi completion Ḡ of G (assumed to be adjoint). They also include as a special case the varieties studied in the first part of this series [L9] (where y = 1 that is, gPg = P ). In this special case a theory of ”character sheaves” on Z was developed in [L9]. In the present paper we extend the theory of character sheaves to a general Z. We now review the content of this paper in more detail. (The numbering of sections continues that of [L9]; we also follow the notation of [L9] .) In Section 8 we introduce a partition of Z similar to that in [L9]; as in [L9], it is based on the combinatorics in Section 2. But whereas in [L9] the combinatorics needed is covered by the results in [B], for the present paper we actually need the slight generalization of [B] given in Section 2. Now, it is not obvious that the partition of Z defined in Section 8 reduces for y = 1 to that in Section 3; this needs an argument that is given in Section 9. In Section 10 we consider the example where G is a general linear group. In Section 11 we define the ”parabolic character sheaves” on Z. As in the case y = 1 (Section 4), we give two definitions for these; one uses the partition in Section 8 and allows us to enumerate the parabolic character sheaves; the other one imitates the definition of character sheaves in [L3]. (The two definitions are equivalent by 11.15 and 11.18.) The theory of character sheaves in Section 11 generalizes that in Section 4 (this is seen from the second definition). A consequence of the coincidence of the two definitions of parabolic character sheaves on Z is that a statement like 0.1(a) (concerning characteristic functions over a finite field) continues to hold in the generality of this paper. In