Klyachko models of p-adic special linear groups

Abstract

We study Klyachko models of ${\rm SL}(n, F)$, where $F$ is a nonarchimedean local field. In particular, using results of Klyachko models for ${\rm GL}(n, F)$ due to Heumos, Rallis, Offen and Sayag, we give statements of existence, uniqueness, and disjointness of Klyachko models for admissible representations of ${\rm SL}(n, F)$, where the uniqueness and disjointness are up to specified conjugacy of the inducing character, and the existence is for unitarizable representations in the case $F$ has characteristic 0. We apply these results to relate the size of an $L$-packet containing a given representation of ${\rm SL}(n, F)$ to the type of its Klyachko model, and we describe when a self-dual unitarizable representation of ${\rm SL}(n, F)$ is orthogonal and when it is symplectic.