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

DOI10.4171/JEMS/1093arXiv1808.02640WikidataQ113691998 ScholiaQ113691998MaRDI QIDQ6305171

Maxim Gurevich

Publication date: 8 August 2018

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 $G{L}_n$ 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-Minguez criterion from the p-adic setting.

Mathematics Subject Classification ID

Ordinary representations and characters (20C15) Quantum groups (quantized function algebras) and their representations (20G42) Representations of Lie and linear algebraic groups over local fields (22E50)

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