JOURNAL OF ALGEBRA 104, 301-309 (1986) Generalized Harish-Chandra Theory for Unipotent Characters of Finite Classical Groups PAUL FONC AND BHAMA SRINIVASAN* TO SANDY GREEN ON HIS 60~~ BIRTHDAY 1. INTRODUCTION The Harish-Chandra theory for a finite group G of Lie type states the following: If p is an irreducible character of G, then there is a parabolic subgroup P of G, a Levi decomposition Lb’ of P, and a cuspidal character II/ of the Levi subgroup such that p is a constituent of the induced character indg($) of the pullback $ of $ to P. Furthermore, the pair (L, $) is uni- quely determined by p up to conjugacy in G. In the case where G is a classical group, that is, a general linear, unitary, symplectic, or orthogonal group over cFq, we generalize this theory relative to an odd prime r different from the defining characteristic. There is an integer e and a polynomial 4(X) of the form x’- 1 or Y’+ 1 with r dividing c$(q) such that the following holds: To each unipotent character p of G corresponds a pair (L, Ic/), where L is a regular subgroup of G of the form a product of a classical group and cyclic tori of order 4(q), Ic, is a unipotent character of L of degree divisible by the full power of r dividing I L: Z(L)l, and p is a constituent of the virtual character Rf(IC/). Moreover, the pair (L, $) is determined by p up to conjugacy in G. The Harish-Chan- dra theory is the case e = I. This work arose in an investigation by the authors of the Brauer r-blocks of G. A similar generalization of the Harish-Chandra theory has been proposed by R. Boyce. Notation. Let G be an algebraic general linear, symplectic, or special orthogonal group defined over Y “,, F a Frobenius endomorphism of G, and G the finite group G’ of F-fixed points of G. Then G is isomorphic to GL(n, q), U(n, q), Sp(2n, q), SO(2n + 1, q), or S0”(2n, q) for some n. A subgroup L of G is regular if L = E” for some F-fixed Levi subgroup ,? of a * This work was supported by NSF Grants MC%8300855 and MCS-8402207. 301 0021-X693/86 $3.00 CopyrIght I( 1986 by Academtc Press, Inc. All rights of reproduction m any form reserved