Generalized Harish-Chandra theory for unipotent characters of finite classical groups

Abstract

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, $) is uniquely determined by p up to conjugacy in G. In the case where G is a classical group, that is, a general linear, unitary, symplectic, or orthogonal group over cFq, we generalize this theory relative to an odd prime r different from the defining characteristic. There is an integer e and a polynomial 4(X) of the form x’1 or Y’+ 1 with r dividing c$(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-Chandra 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.