Smooth representations of involutive algebra groups over non-archimedean local fields

Abstract

An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $\sigma:J\to J$, then $\sigma$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(\sigma)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(\sigma)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(\sigma)$ where $H$ is an $\sigma$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$.