Langlands parameters, functoriality and Hecke algebras

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-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 $\eta^* \pi$. 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 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.