Langlands parameters, functoriality and Hecke algebras

Maarten Solleveld

Vol. 304 (2020), No. 1, 209–302

DOI: 10.2140/pjm.2020.304.209

Abstract

Let $G$ and $\tilde{G}$ be reductive groups over a local field $F$. Let $\eta :\tilde{G}\to G$ be an $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible $G$-representations $\pi$ along $\eta$. Following Borel, Adler–Korman and Xu, we pose a conjecture on the decomposition of the pullback ${\eta }^{\ast }\pi$. 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 ${GL}_n,{SL}_n,{PGL}_n$.

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