On compatibility in restriction of Arthur's conjecture for $R$-groups

TL;DR

This paper investigates how Arthur's conjecture for R-groups behaves when restricting discrete series representations from a p-adic group to its subgroup with the same derived group, highlighting conditional compatibility and potential verification methods.

Contribution

It introduces a framework for understanding the compatibility of Arthur's conjecture for R-groups under restriction, proposing a uniform approach for verification.

Findings

Compatibility holds conditionally under the local Langlands correspondence.

Application to several cases demonstrates the framework's utility.

Suggests a uniform method for verifying Arthur's conjecture for R-groups.

Abstract

We study the compatibility of Arthur's conjecture for $R$-groups in the restriction of discrete series representations from Levi subgroups of a $p$-adic group to those of its closed subgroup having the same derived group. The compatibility is conditional as it holds under the conjectural local Langlands correspondence in general. This work is applied to several cases and further suggests a uniform way to verify the Arthur's conjecture for $R$-groups in the setting.