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 mod \(p\) cohomological automorphic forms on \(\mathbf{G}\) and \(\mathbf{G}^\sigma\). We describe this correspondence in the global and local settings.