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.)

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Last updated: 03 Feb 2023