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

Sécherre, Vincent and Stevens, Shaun (2019) Towards an explicit local Jacquet-Langlands correspondence beyond the cuspidal case. Compositio Mathematica, 155 (10). pp. 1853-1887. ISSN 0010-437X

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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 remar- kable 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), n > 0. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence π -> JL(π) 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. Mínguez 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 π and JL(π) 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 π and JL(π) are both cuspidal.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: LivePure Connector
Date Deposited: 12 Apr 2019 08:30
Last Modified: 15 Nov 2020 00:59
URI: https://ueaeprints.uea.ac.uk/id/eprint/70543
DOI: 10.1112/S0010437X19007486

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