Quotient branching laws and unitary Gan-Gross-Prasad relevance for general linear groups

Abstract

This article addresses the quotient branching problem for the pair $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$ over a non-archimedean local field $F$. We present two primary contributions. First, we explicate Chan's recent general solution by providing a practical, step-by-step combinatorial algorithm to determine whether the space $\mathrm{Hom}_{\mathrm{GL}_n(F)}(\pi, \pi')$ is non-zero for any irreducible smooth representations $\pi$ and $\pi'$. This algorithm is designed to be verifiable by hand, given the representations' Langlands or Zelevinsky parameters. Second, we specialize to the unitary case, where we establish a unifying equivalence: a pair of irreducible unitary representations satisfies Chan's generalized GGP relevance criterion if and only if it satisfies a natural extension of the original Gan--Gross--Prasad relevance condition. This result resolves a previous inconsistency in the literature, corrects and extends prior work by Gurevich, and provides a complete, explicit criterion for unitary branching laws.