Klyachko models of p-adic special linear groups

TL;DR

This paper investigates Klyachko models for ${ m SL}(n, F)$ over nonarchimedean local fields, establishing existence, uniqueness, and disjointness results, and relating these models to $L$-packets and representation types.

Contribution

It extends Klyachko model theory from ${ m GL}(n, F)$ to ${ m SL}(n, F)$, providing new existence and classification results for admissible representations.

Findings

Existence of Klyachko models for unitarizable representations in characteristic zero.

Uniqueness and disjointness of models up to conjugacy of inducing characters.

Relation between $L$-packet size and Klyachko model type.

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.