Combinatorics and Algebra Seminar-Completely Reducible Automorphisms of Quasi-Reductive Groups

Tuesday, April 21, 2026 4:00 pm - 5:00 pm

Title: Completely Reducible Automorphisms of Quasi-Reductive Groups
Speaker: Joshua Lansky, American U
Date and time: Tuesday, April 21, 4–5 pm
Place: Rome 206

Abstract: Let $k$ be a field, and suppose that $\Gamma$ is a smooth $k$ -group acting on a smooth, connected $k$ -group $\widetilde G$ . Let $G$ denote the maximal smooth, connected subgroup of the group of $\Gamma$ -fixed points in $\widetilde G$ . In joint work with Adler and Spice, when $\widetilde G$ is reductive and $\Gamma$ stabilizes a Borel pair, we show that, under fairly general conditions, $G$ is a reductive $k$ -group and that the spherical (and affine, if $k$ is quasi-local) buildings of $\widetilde G$ and $G$ behave nicely with respect to the action of $\Gamma$ . We motivate the broader class of so-called quasi-reductive groups and the notion of a completely reducible action of $\Gamma$ on such a group $\widetilde G$ in order to discuss broad generalizations of some of the preceding results.