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We show how the modular representation theory of inner forms of generallinear groups over a non-Archimedean local field can be brought to bear on thecomplex theory in a remarkable way. Let F be a non-Archimedean locally compactfield of residue characteristic p, and let G be an inner form of the generallinear group GL(n,F). We consider the problem of describing explicitly thelocal Jacquet--Langlands correspondence between the complex discrete seriesrepresentations of G and GL(n,F), in terms of type theory. We show that thecongruence properties of the local Jacquet--Langlands correspondence exhibitedby A. M\'inguez and the first named author give information about the explicitdescription of this correspondence. We prove that the problem of the invarianceof the endo-class by the Jacquet--Langlands correspondence can be reduced tothe case where the representations $\pi$ and its Jacquet--Langlands transferJL($\pi$) are both cuspidal with torsion number 1. We also give an explicitdescription of the Jacquet--Langlands correspondence for all essentially tamediscrete series representations of G, up to an unramified twist, in terms ofadmissible pairs, generalizing previous results by Bushnell and Henniart. Inpositive depth, our results are the first beyond the case where $\pi$ andJL($\pi$) are both cuspidal.

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