Unipotent Ideals for Spin and Exceptional Groups

TL;DR

This paper extends the theory of unipotent representations to spin and exceptional groups, providing formulas for their infinitesimal characters, proving maximality of unipotent ideals, and establishing unitarity for certain real forms.

Contribution

It develops combinatorial formulas for unipotent representations of spin and exceptional groups and proves the maximality and unitarity properties of their unipotent ideals.

Findings

Formulas for infinitesimal characters of unipotent ideals in spin and exceptional groups.

Proof that all unipotent ideals are maximal.

Demonstration of unitarity for unipotent representations of real forms of exceptional groups.

Abstract

In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of Barbasch-Vogan and form the (conjectural) building blocks of the unitary dual. In arXiv:2108.03453 we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal characters of all unipotent ideals. Using these formulas, we prove an old conjecture of Vogan: all unipotent ideals are maximal. For $G$ a real reductive Lie group (not necessarily complex), we introduce the notion of a unipotent representation attached to a rigid nilpotent orbit (in the complexified Lie algebra of $G$). Like their complex group counterparts, these representations form the (conjectural) building blocks of the unitary dual. Using the atlas software (and the work of Adams-Miller-van Leeuwen-Vogan) we show that if $G$ is a real form of a simple group of exceptional type, all such representations are unitary.