On Arthur representations and the unitary dual

TL;DR

The paper proposes a conjectural description of the unitary dual $\Pi_u(G)$ for connected reductive groups over non-Archimedean local fields of characteristic zero in terms of Arthur representations, introducing the candidate set $\Pi_{\overline{A}}^{\lim}(G)$ built from Arthur data via unitary constructions. It provides an explicit algorithm (Algorithm Abar) that generates $\Pi_{\overline{A}}^{\lim}(G)$ for classical groups and proves the conjecture in the exceptional type $G_2$, all under the local Arthur framework. The authors develop two new algorithms to decide whether a representation is of Arthur type from enhanced Langlands parameters, and they establish an inductive, corank-based classification of Arthur representations, aided by extended multi-segments and a suite of operators. They also present a broad array of evidence, including compatibility with generic and unramified unitary duals, corank verifications up to corank 4 for classical groups, and a full verification for $G_2$, along with consequences such as the preservation of the unitary dual under Aubert-Zelevinsky duality. Collectively, the work lays groundwork for algorithmic determination of the unitary dual and clarifies the structural role of Arthur packets in the non-Archimedean setting.

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 $G_2$.

Figures (7)

• Figure 1: Unitary subquotients of $\Pi_{x,y}$
• Figure 2: Regions where $\Pi_{x,y}$ is irreducible and unitary
• Figure 3: Reducibility hyperplanes of $\rho\vert\cdot\vert^x\rtimes\sigma$
• Figure 4: 2-dimensional regions of unitarity for $\Pi_{x,y}$
• Figure 5: Reducibility hyperplanes of $u_\rho(2,1)\vert\cdot\vert^t\rtimes\sigma$

Theorems & Definitions (106)

• Conjecture 1.1: Unitary Dual
• Theorem 1.2
• Theorem 1.3: LLW
• Corollary 1.4: Corollary \ref{['cor implication of 1.1']} and Remark \ref{['rmk-general-classical']}
• Conjecture 1.5: HJLLZ24
• Theorem 1.6: AM25HLL25
• Theorem 2.2: Art13
• Theorem 2.3: Moe11a, Xu17a
• Definition 2.4
• Proposition 2.5