On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture

Abstract

We prove the first direction of a recently posed conjecture by Gan, Gross and Prasad, which predicts branching laws that govern restriction from $p$-adic $G{L}_n$ to $G{L}_{n-1}$ of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur–Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid–Mínguez criterion from the $p$-adic setting.