Restrictions of representations of classical groups: examples

TL;DR

This paper investigates restriction problems in the representation theory of classical groups, proposing conjectures based on Langlands parameters and verifying some cases using advanced methods.

Contribution

It formulates precise conjectures for restriction problems in classical groups and verifies several low-rank cases using diverse mathematical techniques.

Findings

Conjectures relate restriction problems to Langlands parameters and symplectic root numbers.

Verification of conjectures in low-rank cases using base change, theta correspondence, and global methods.

Provides evidence supporting the proposed conjectural framework.

Abstract

In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands-Vogan parameterization of irreducible representations, we formulated precise conjectures for the solutions of these restriction problems. In the local case, our conjectural answer is given in terms of Langlands parameters and certain natural symplectic root numbers associated to them. In the global case, the conjectural answer is expressed in terms of the central critical value or derivative of a global $L$-function. In this paper we verify several of these conjectures in certain low rank cases, using methods of base change, theta correspondence, and global arguments.