Smooth representations of ${GL}_m(D)$. V: Endo-classes

Abstract

Let $\mathrm{F}$ be a locally compact nonarchimedean local field. In this article, we extend to any inner form of ${\mathrm{GL}}_n$ over $\mathrm{F}$, with $n⩾1$, the notion of endo-class introduced by Bush­nell and Henniart for ${\mathrm{GL}}_n(\mathrm{F})$. We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of ${\mathrm{GL}}_n(\mathrm{F})$ an endo-class over $\mathrm{F}$. We conjecture that this endo-class is invariant under the local Jacquet–Langlands correspondence.