Branching laws for Classical Groups: the non-tempered case

TL;DR

This paper extends the GGP conjectures to Arthur packets, especially non-generic cases, introducing relevant A-parameters that govern branching laws for classical groups over local and global fields.

Contribution

It generalizes the GGP conjectures to non-tempered Arthur packets and introduces relevant A-parameters for classical groups and $GL_n$ in branching problems.

Findings

Formulation of the relevant A-parameters concept.

Extension of branching laws to non-tempered Arthur packets.

Unified approach for local and global fields.

Abstract

This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models.