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

Contact: Peter Scholze (scholze@mpim-bonn.mpg.de)

Let $k$ be a nonarchimedian local field of residual characteristic $p$, $\widetilde{G}$ a connected reductive $k$-group, $\Gamma$ a finite group of automorphisms of $\widetilde{G}$, and $G$ the connected part of the group of $\Gamma$-fixed points of $\widetilde{G}$.
If one assumes that the order of $\Gamma$ is coprime to $p$, then Prasad-Yu and Kaletha-Prasad show, roughly speaking, that $G$ is reductive, the building of $G$ embeds in the set of $\Gamma$-fixed points of the building of $\widetilde{G}$, similarly for spherical buildings, and similarly for reductive quotients of parahoric subgroups.

Motivated by the desire for a more explicit understanding of base change and other liftings, we prove similar statements under a different hypothesis on $\Gamma$. Our hypothesis does not imply that of Kaletha-Prasad-Yu, 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.)