The duality operation in the character ring of a finite Chevalley group

Abstract

It is possible (as in Such a group arises as the fixed points under a Frobenius map of a connected reductive algebraic group, defined over a finite field This paper contains statements of several general properties of the duality map f - f * and two related operations (see 2 and 4). The duality map f - f * generalizes the construction in [2] of the Steinberg character, and interacts well with the organization of the characters from the point of view of cuspidal characters ( 6). It is hoped that there is also a useful interaction with the Deligne-Lusztig virtual characters R^O. Partial results have been obtained in this direction ( 5). Detailed proofs will appear elsewhere.