$R$-groups and geometric structure in the representation theory of $\mathrm{SL}(N)$

Abstract

Let $F$ be a nonarchimedean local field of characteristic zero and let $G=\mathrm{SL}(N)=\mathrm{SL}(N,F)$. This article is devoted to studying the influence of the elliptic representations of $\mathrm{SL}(N)$ on the K-theory. We provide full arithmetic details. This study reveals an intricate geometric structure. One point of interest is that the $R$-group is realized as an isotropy group. Our results illustrate, in a special case, part (3) of the recent conjecture in [Aubert, Baum, Plymen (2007)].