Smooth representations of GL(m,D), V: Endo-classes

TL;DR

This paper extends the concept of endo-classes to inner forms of GL(n) over nonarchimedean local fields, analyzing their properties and invariance under transfer and conjectural correspondences.

Contribution

It generalizes Bushnell and Henniart's endo-class notion to all inner forms of GL(n) over local fields and studies their transfer and invariance properties.

Findings

Endo-classes are associated to discrete series representations of inner forms.

Intertwining relations of simple characters are preserved under transfer.

Conjecture: Endo-classes are invariant under the local Jacquet-Langlands correspondence.

Abstract

Let F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of GL(n) over F the notion of endo-class introduced by Bushnell and Henniart for GL(n,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 GL(n,F) an endo-class over F. We conjecture that this endo-class is invariant under the local Jacquet-Langlands correspondence.