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

TL;DR

This paper proves a significant part of the Gan-Gross-Prasad conjecture for unitarizable representations of p-adic GL(n), using quantum algebra techniques to analyze branching laws and restriction phenomena.

Contribution

It extends the Gan-Gross-Prasad conjecture to all unitarizable representations and confirms it when one representation is generic, employing quantum affine algebra methods.

Findings

Confirmed the conjecture for generic cases.

Established a combinatorial relation for unitarizable representations.

Connected p-adic representation theory with quantum affine algebras.

Abstract

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ 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-Minguez criterion from the p-adic setting.