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Abstract
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Let
and
be reductive groups
over a local field
.
Let
be an
-homomorphism with
commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible
-representations
along
. Following
Borel, Adler–Korman and Xu, we pose a conjecture on the decomposition of the pullback
.
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
nonarchimedean)
and inner twists of
.
Our main techniques involve Hecke algebras associated to Langlands
parameters. We also prove a version of the pullback/functoriality conjecture for
those.
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Keywords
reductive groups, representations, local Langlands
correspondence, functoriality, Hecke algebras
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Mathematical Subject Classification 2010
Primary: 20G25
Secondary: 11S37, 20C08
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Milestones
Received: 14 November 2018
Accepted: 13 July 2019
Published: 18 January 2020
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