On Arthur representations and the unitary dual

TL;DR

This paper introduces a conjecture linking Arthur representations to the structure of the unitary dual for reductive groups over non-Archimedean fields, supported by explicit algorithms and verified cases.

Contribution

It proposes a new conjecture for the unitary dual structure, providing an explicit candidate set derived from Arthur representations and verifying it for specific groups.

Findings

Candidate set aligns with known unitary duals

Algorithm successfully generates candidate sets for classical groups

Conjecture verified for the G2 exceptional group

Abstract

In this paper, we propose a new conjecture describing the structure of the unitary dual in terms of Arthur representations for connected reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. This conjecture provides a candidate set for the unitary dual, constructed from Arthur representations. For classical groups, we develop an explicit algorithm to generate this candidate set. Evidence for its exhaustiveness includes compatibility with the known generic unitary dual, unramified unitary dual, and low-corank representations. As further support, we verify the conjecture for the unitary dual of the exceptional group of type $G_2$.