The Duality Operation in the Character Ring of a Finite Chevalley Group by Dean Alvis

It is possible (as in [4] ) to define a duality operation f —• f * in the ring of virtual characters of an arbitrary finite group with a split (B, 7V)-pair of characteristic p. Such a group arises as the fixed points under a Frobenius map of a connected reductive algebraic group, defined over a finite field [1]. 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.