Arthur's Conjectures and the Orbit Method for Real Reductive Groups

TL;DR

This paper reviews Arthur's conjectures and the Langlands classification for real reductive groups, introduces a conjectural Orbit Method for classifying irreducible unitary representations, and explores its connections and potential generalizations.

Contribution

It defines the Orbit Method for complex groups, relates it to Arthur's conjectures, and proposes a possible extension to all real groups.

Findings

Orbit Method for complex groups is defined and related to Arthur's conjectures.

A duality map connects the Orbit Method with Arthur's conjectures.

Sketches a potential generalization of the Orbit Method for all real groups.

Abstract

The first half of this article is expository -- I will review, with examples, the main statements of the Langlands classification and Arthur's conjectures for real reductive groups as formulated by Adams, Barbasch, and Vogan. In the second half, I will turn my attention to the Orbit Method, a conjectural scheme for classifying irreducible unitary representations of a real reductive group. I will give a definition of the Orbit Method in the case when the group is complex. The main input is the theory of unipotent ideals and Harish-Chandra bimodules, developed in arXiv:2108.03453. I will show that the Orbit Method I define is related to Arthur's conjectures via a natural duality map. Finally, I will sketch a possible generalization of this Orbit Method for arbitrary real groups.