Dual $R$-groups of the inner forms of $\mathrm{SL}(N)$

Kuok Fai Chao and Wen-Wei Li

Vol. 267 (2014), No. 1, 35–90

DOI: 10.2140/pjm.2014.267.35

Abstract

We study the Knapp–Stein $R$-groups of the inner forms of $SL\left(N\right)$ over a nonarchimedean local field of characteristic zero, by using a restriction from the inner forms of $GL\left(N\right)$. As conjectured by Arthur, these $R$-groups are then shown to be naturally isomorphic to their dual avatars defined in terms of L-parameters. The $2$-cocycles attached to $R$-groups can be described as well. The proofs are based on the results of K. Hiraga and H. Saito. We also construct examples to illustrate some new phenomena which do not occur in the case of $SL\left(N\right)$ or classical groups.

Keywords

R-group, local Langlands correspondence, intertwining operator

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 11F70

Milestones

Received: 2 December 2012

Accepted: 5 March 2013

Published: 22 December 2013