Functoriality, Smith theory, and the Brauer homomorphism

Abstract

If $\sigma$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between $\mathrm{mod}p$ cohomological automorphic forms on $\mathbf{G}$ and $\mathbf{G}^{\sigma}$. We describe this correspondence in the global and local settings.