On unipotent blocks and their ordinary characters

Abstract

There is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the Harish-Chandra theory for those groups (see [S, 5.1; FS2; P2, 3.4; H, 6.1.10]). An e-generalized Harish-Chandra theory (for e a positive integer) is introduced in [BMM, I.A]. The context is as follows. Let G be a connected reductive group defined over lFq with associated Frobenius endomorphism F, let e > 1 an integer. When L is an F-stable Levi subgroup of G, one denotes by R~ and *RE the Lusztig functor and its adjoint. A unipotent e-pair (L, L) is defined by an e-split Levi subgroup L of G and a unipotent character Z~d'(L ~', 1) . One writes (LI, Z1) < (L2, J(2) if and only if L1 c L 2 and (Zl, *~,L2, \ , "L, 1.2/L 4= 0 (see 3.l). The notion of unipotent e-cuspidal pairs and the resulting partition of g(G v, 1) are defined as in the classical case (corresponding to e = 1). If f is a prime not dividing q and e is the order of q mod f, it seems natural in connection with blocks, to ask if the partition of the unipotent characters 8 (G v, l) resulting from "e-theory" coincides with f-blocks. The blocks of G v which are defined by unipotent characters are called unipoteut f-blocks. We prove