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.)