Unipotent ideals for spin and exceptional groups

In the monograph [27], we define the notion of a unipotent representation of a complex reductive group. The representa-tions we define include, as a proper subset, all special unipo-tent representations in the sense of [4] and form the (conjec-tural) building blocks of the unitary dual. In [27] we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this pa-per, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesi-mal 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 representa-tion 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 [2]) we show that if G is a real form of a simple group of excep-tional type, all such representations are unitary.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).