  Let $F$ be a nonarchimedean local field of characteristic zero and let SL(N)
= SL(N,F). This article is devoted to studying the influence of the elliptic
representations of 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 [3].
