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

Part of: Lie groups

Published online by Cambridge University Press:  27 August 2019

Vincent Sécherre  and
Shaun Stevens
Vincent Sécherre
Affiliation:
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France email vincent.secherre@math.uvsq.fr
Shaun Stevens
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK email Shaun.Stevens@uea.ac.uk
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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 remarkable way. Let $\text{F}$ be a non-Archimedean locally compact field of residue characteristic $p$, and let $\text{G}$ be an inner form of the general linear group $\text{GL}_{n}(\text{F})$ for $n\geqslant 1$. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence $\unicode[STIX]{x1D70B}\mapsto _{\text{JL}}\unicode[STIX]{x1D70B}$ between the complex discrete series representations of $\text{G}$ and $\text{GL}_{n}(\text{F})$, in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first 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 $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ 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 $\text{G}$, 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 $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal.

Keywords

Jacquet–Langlands correspondenceℓ-blockscongruences mod ℓendo-classadmissible pairrectifier

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 10 , October 2019 , pp. 1853 - 1887
Copyright
© The Authors 2019 

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