Vol. 304, No. 1, 2020

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Langlands parameters, functoriality and Hecke algebras

Maarten Solleveld

Vol. 304 (2020), No. 1, 209–302
Abstract

Let G and G˜ be reductive groups over a local field F. Let η : G˜ G be an F-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible G-representations π along η. Following Borel, Adler–Korman and Xu, we pose a conjecture on the decomposition of the pullback ηπ. It is formulated in terms of enhanced Langlands parameters and includes multiplicities. This can be regarded as a functoriality property of the local Langlands correspondence.

We prove this conjecture for three classes: principal series representations of split groups (over nonarchimedean local fields), unipotent representations (also with F nonarchimedean) and inner twists of GLn,SLn,PGLn.

Our main techniques involve Hecke algebras associated to Langlands parameters. We also prove a version of the pullback/functoriality conjecture for those.

Keywords
reductive groups, representations, local Langlands correspondence, functoriality, Hecke algebras
Mathematical Subject Classification 2010
Primary: 20G25
Secondary: 11S37, 20C08
Milestones
Received: 14 November 2018
Accepted: 13 July 2019
Published: 18 January 2020
Authors
Maarten Solleveld
IMAPP
Radboud Universiteit Nijmegen
Nijmegen
Netherlands