R-groups and parameters

TL;DR

This paper proves the isomorphism between the Knapp-Stein and Arthur R-groups for classical groups, specifically for discrete series representations, advancing understanding in the local Langlands correspondence.

Contribution

It establishes the isomorphism of R-groups for classical groups and discrete series, with a focus on unitary groups under mild assumptions, linking representation structure and Langlands parameters.

Findings

Isomorphism of R-groups for classical groups proven

Results hold for discrete series representations

Partial results for unitary groups under mild assumptions

Abstract

For classical groups we show the isomorphism of the Knapp-Stein $R$-group, which describes the structure of parabolically induced representations, and the Arthur $R$-group of the parameter associated to the inducing representation by the local Langlands conjecture. We do this in the case of inducing from discrete series representations. In the case of unitary groups we show this isomorphism under a mild assumption on the parameter, which we show holds in at least half the cases.