Weak unipotence and Langlands duality

TL;DR

The paper addresses weak unipotence for primitive ideals in $\mathcal{U}\mathfrak{g}$ by introducing mild unipotence as a practical sufficient condition. It proves mild unipotence for two principal classes—$q$-unipotent infinitesimal characters (McGovern) and unipotent ideals attached to nilpotent orbit covers (Losev–Mason–Brown–Matvieievskyi)—in classical types, and extends to birationally rigid covers in exceptional types using atlas-guided methods. The approach intertwines Kazhdan–Lusztig/Barbasch–Vogan cell theory, coherent families, and Langlands duality, leveraging dualities between type $D$ and type $C$ (and metaplectic analogs) to reduce to special unipotent situations. Consequently, a large class of natural unipotent-like ideals are shown to be weakly unipotent, strengthening connections to unitary representation theory and providing a framework for further duality-driven analyses across classical and exceptional groups.

Abstract

Weak unipotence of primitive ideals is a crucial property in the study of unitary representations of reductive groups. We establish a sufficient condition, referred to as mild unipotence, which guarantees weak unipotence and is more accessible in practice. We establish mild unipotence for both the $q$-unipotent ideals defined by McGovern and unipotent ideals attached to nilpotent orbit covers defined by Losev-Mason-Brown-Matvieievskyi (arXiv:2108.03453 [math.RT]). Our proof is conceptual and uses the bijection between special orbits in type $D$ and metaplectic special orbits in type $C$ found by Barbasch-Ma-Sun-Zhu (arXiv:2010.16089 [math.RT]) in an essential way.

Theorems & Definitions (60)

• Theorem 1.1
• Definition 2.1: c.f. Vogan:unitarizability
• Proposition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof
• Corollary 2.6
• proof