Functoriality, Smith theory, and the Brauer homomorphism
  If $\sigma$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between mod $p$ cohomological automorphic forms on $\mathbf{G}$ and $\mathbf{G}^\sigma$. We describe this correspondence in the global and local settings. 
David Treumann and Akshay Venkatesh
math.NT