Duality for generalized Gan-Gross-Prasad relevant pairs for $p$-adic $\mathrm{GL}_n$

TL;DR

This paper introduces a new concept called generalized GGP relevant pairs for $p$-adic $ ext{GL}_n$, establishing a duality that relates derivatives and integrals in the context of branching laws.

Contribution

It formulates the notion of generalized GGP relevant pairs and proves a duality principle that aligns with dual restrictions in branching laws for $p$-adic $ ext{GL}_n$.

Findings

Defines generalized GGP relevant pairs for $p$-adic $ ext{GL}_n$

Establishes a duality compatible with branching law restrictions

Provides representation-theoretic and combinatorial insights

Abstract

The main goal of this article is to formulate a notion, called a generalized GGP relevant pair, governing the quotient branching law for $p$-adic general linear groups. Such notion relies on a commutation relation between derivatives (from Jacquet functors) and integrals (from parabolic inductions), for which we provide both representation-theoretic and combinatorial perspectives. Our main result proves a duality on those relevant pairs, which is compatible with a dual restriction in branching law.