Lifting Representations of Finite Reductive Groups I : Semisimple Conjugacy Classes : semisimple conjugacy classes

Suppose that (G) over tilde is a connected reductive group defined over a field k, and Gamma is a finite group acting via k-automorphisms of (G) over tilde satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of Gamma-fixed points in (G) over tilde is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair ((G) over tilde, Gamma), and consider any group G satisfying the axioms. If both (G) over tilde and G are k-quasisplit, then we can consider their duals (G) over tilde* and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G* (k) to the analogous set for (G) over tilde* (k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (G) over tilde (k), one obtains a mapping of such packets.