The Unitarity of Arthur Packets for Real Reductive Groups

Abstract: Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parameters. A longstanding conjecture, originally due to Arthur, is that all Arthur packets consist of $\textit{unitary}$ representations. In this paper, we prove this conjecture in general. The main new idea is a `Jordan decomposition' for Arthur packets: a canonical two-step process for realizing an arbitrary Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a certain Levi subgroup. This process is analogous to the decomposition of an element of a complex algebraic group as a (unique) commuting product of elliptic, hyperbolic, and unipotent parts. Using our Jordan decomposition, we reduce the question of unitarity to the case of unipotent Arthur packets, where the answer is already known (by work of Adams-Arancibia-Mezo, Adams-van Leeuwen-Miller-Vogan, Arthur, Barbasch, Barbasch-Ma-Sun-Zhu, and Davis-Mason-Brown). As an application of the same methods, we also give a proof of Jiang's conjecture for real reductive groups, which gives an upper bound on the wavefront sets of the members of an Arthur packet in terms of the Barbasch-Vogan dual of the Arthur $SL_2(\mathbb{C})$.