Certain Unitary Langlands-Vogan Parameters for Special Orthogonal Groups I

TL;DR

This paper proves the unitarity of certain Langlands-Vogan parameters for special orthogonal groups, advancing understanding of Arthur's conjecture in this context.

Contribution

It establishes unitarity for a specific class of parameters in Arthur's conjecture for special orthogonal groups, a case previously unverified.

Findings

Confirmed unitarity for a class of Langlands-Vogan parameters

Supported Arthur's conjecture in a basic case

Enhanced understanding of representation theory for special orthogonal groups

Abstract

A Langlands parameter, in the Langlands dual group, can be decomposed into a product of a tempered parameter and a positive quasi-character. Fixing a tempered parameter, Arthur conjectured that positive quasi-characters corresponding to certain weighted Dynkin diagrams for the centralizer of the tempered parameter will yield unitary representations. In this paper, we treat one basic case in Arthur's conjecture for the special orthogonal groups. We establish the unitarity for a class of Langlands-Vogan parameters in Arthur's packet.