Lifting representations of finite reductive groups I: Semisimple conjugacy classes

Suppose that \(\tilde{G}\) is a connected reductive group defined over a field \(k\), and \(\Gamma\) is a finite group acting via \(k\)-automorphisms of \(\tilde{G}\) satisfying a certain quasi-semisimplicity condition. Then the connected part of the group of \(\Gamma\)-fixed points in \(\tilde{G}\) is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair \((\tilde{G},\Gamma)\), and consider any group \(G\), not just the \(\Gamma\)-fixed points of \(\tilde{G}\), satisfying the axioms. (In fact, the axioms do not require \(\Gamma\) to act on all of \(\tilde{G}\).) If both \(\tilde{G}\) and \(G\) are \(k\)-quasisplit, then we can consider their duals \(\tilde{G}^*\) and \(G^*\). We show the existence of and give an explicit formula for a natural map from semisimple stable conjugacy classes in \(G^*(k)\) to those in \(\tilde{G}^*(k)\). If \(k\) is finite, then our groups are automatically quasisplit, and our result specializes to give a map from semisimple conjugacy classes in \(G^*(k)\) to those in \(\tilde{G}^*(k)\). Since such classes parametrize packets of irreducible representations of \(G(k)\) and \(\tilde{G}(k)\), one obtains a mapping of such packets.