Finite-group actions on reductive groups and buildings II: the unauthorized sequel

Abstract:
Let $k$ be a nonarchimedian local field, $\widetilde{G}$ a connected
reductive $k$-group, $\Gamma$ a finite group of automorphisms of
$\widetilde{G}$, and $G:= (\widetilde{G}^\Gamma)^\circ$ the connected part
of the group of $\Gamma$-fixed points of $\widetilde{G}$.
The first half of my talk will concern motivation: a desire for a more
explicit understanding of base change and other liftings of
representations.  Toward this end, we adapt some results of
Kaletha-Prasad-Yu.  Namely, if one assumes that the residual characteristic
of $k$ does not divide the order of $\Gamma$, then they show, roughly
speaking, that $G$ is reductive, the building $\mathcal{B}(G)$ of $G$
embeds in the set of $\Gamma$-fixed points of $\mathcal{B}(\widetilde{G})$,
and similarly for reductive quotients of parahoric subgroups.

We prove similar statements, but under a different hypothesis on $\Gamma$.
Our hypothesis does not imply that of K-P-Y, nor vice versa. I will include
some comments on how to resolve such a totally unacceptable situation.

(This is joint work with Joshua Lansky and Loren Spice.)