Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups

TL;DR

This paper proves the isomorphism between two types of R-groups for p-adic inner forms of quasi-split special unitary groups, confirming a conjecture and exploring invariance properties within L-packets.

Contribution

It establishes Arthur's conjecture for these groups and investigates R-group invariance, extending known results from quasi-split to inner forms.

Findings

Proved the isomorphism between Knapp-Stein and Langlands-Arthur R-groups.

Demonstrated invariance of R-groups within L-packets and between inner forms.

Extended results from quasi-split groups to their non quasi-split inner forms.

Abstract

We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non quasi-split inner forms.