Some branching laws for symmetric spaces

TL;DR

This paper investigates branching laws for unitary symmetric spaces, providing new restriction formulas for discrete series representations and advancing the understanding of Gan-Gross-Prasad conjectures in this context.

Contribution

It formulates and proves natural relative branching laws for unitary symmetric spaces, extending previous work and contributing to the Gan-Gross-Prasad conjecture framework.

Findings

Established branching laws for specific symmetric spaces.

Proved analogues of conjectures related to period integrals.

Connected representation theory with conjectural frameworks.

Abstract

In this paper we consider the unitary symmetric spaces of the form X=U(p,q)/U(1)U(p,q-1) and their discrete series representations. Inspired by the work of A.Venkatesh and Y.Sekellarides on L-groups of p-adic spherical spaces we formulate and prove natural relative branching laws for the restriction to smaller subgroups of the same type and corresponding unitary spaces.We think of this as steps to formulation and proving Gan Gross Prasad conjectures for unitary spaces. Using period integral and some results of T.Kobayashi we prove an analogue of thesis conjectures.