Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

TL;DR

This paper explores restriction problems of classical group representations, connecting root numbers and L-values with new conjectures and results in number theory and representation theory.

Contribution

It extends previous work on restriction problems to broader groups, formulates new conjectures involving root numbers and L-values, and proves several related results.

Findings

Formulated conjectures linking root numbers to restriction problems.

Proved new results in number theory related to L-values.

Extended the scope of restriction problems to classical and metaplectic groups.

Abstract

We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.