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.