#### Vol. 304, No. 1, 2020

 Recent Issues Vol. 311: 1 Vol. 310: 1  2 Vol. 309: 1  2 Vol. 308: 1  2 Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
Langlands parameters, functoriality and Hecke algebras

### Maarten Solleveld

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

Let $G$ and $\stackrel{˜}{G}$ be reductive groups over a local field $F$. Let $\eta :\stackrel{˜}{G}\to G$ 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