Towards an explicit local Jacquet-Langlands correspondence beyond the cuspidal case

TL;DR

This paper advances the explicit understanding of the local Jacquet-Langlands correspondence for inner forms of GL(n) over non-Archimedean fields, extending results beyond the cuspidal case using modular representation theory.

Contribution

It provides an explicit description of the Jacquet-Langlands correspondence for all essentially tame discrete series representations, including positive depth cases, using type theory and congruence properties.

Findings

Reduced invariance problem to cuspidal case with torsion number 1.

Explicit description for all essentially tame discrete series representations.

First positive depth results beyond the cuspidal case.

Abstract

We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands correspondence exhibited by A. M\'inguez and the first named author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet--Langlands correspondence can be reduced to the case where the representations $\pi$ and its Jacquet--Langlands transfer JL($\pi$) are both cuspidal with torsion number 1. We also give an explicit description of the Jacquet--Langlands correspondence for all essentially tame discrete series representations of G, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where $\pi$ and JL($\pi$) are both cuspidal.