Langlands parameters, functoriality and Hecke algebras

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $η: \tilde G \to G$ be a $F$-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 $η^* π$. 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 representation of split groups (over non-archimedean local fields), unipotent representations (also with $F$ non-archimedean) 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.

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