On the complementary Arthur representations and unitary dual for p-adic classical groups

TL;DR

The paper delivers an explicit characterization of complementary Arthur representations for split $p$-adic classical groups $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ by expressing Arthur packets in terms of parabolic inductions of generalized Speh blocks $u_{\rho}(a,b)$ and Arthur-type constituents. The core methodology blends Arthur’s local intertwining relation with Mui\'c–Tadić non-unitarity criteria and a novel extended $\mathbb{Z}$-segment combinatorics to control reducibility and unitarity, culminating in a full proof of the conjectured unitary-closure description $\Pi_{\overline{A}}(G)=\Pi_{\overline{A+,u}}(G)=\Pi_u(G)$ for these groups. A key technical innovation is the introduction of extended multi-segments and the NV-criterion, which organize the local Arthur packets and enable deconstruction of unitary inductions into tractable summands. The work yields concrete constraints on local components of self-dual cuspidal automorphic representations of $\mathrm{GL}_N$, notably for $N=2,3$, thereby impacting the understanding of automorphic spectra without assuming Ramanujan. Overall, the results solidify a path to building the unitary dual from Arthur-type data for classical groups, with broad arithmetic applications.

Abstract

In [HJLLZ24], we proposed a new conjecture on the structure of the unitary dual of connected reductive groups over non-Archimedean local fields of characteristic zero based on their Arthur representations and verified it for all the known cases on the unitary dual problem. One step towards this conjecture involves the question whether certain complementary Arthur representations are unitary. In this paper, we give an explicit characterization of the complementary Arthur representations for symplectic and split odd special orthogonal groups. As applications, we obtain interesting constraints on local components of irreducible self-dual cuspidal automorphic representations of GL(N), especially when N=2,3.

Theorems & Definitions (67)

• Conjecture 1.1: HJLLZ24
• Conjecture 1.2: Complementary Arthur Representations, HJLLZ24
• Theorem 1.3: Theorem \ref{['thm main']}
• Corollary 1.4
• Theorem 1.5: Theorem \ref{['thm Ramanujan']}
• Remark 1.6
• Theorem 1.7: HJLLZ24
• Theorem 2.1: Moe11b
• Remark 2.2
• Lemma 3.1: MT11