Weak unipotence and Langlands duality

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.