Let G be a real reductive Lie group. One of the most basic unsolved problems in representation theory and abstract Harmonic analysis is the classification of the set Irr u pGq of irreducible unitary G-representations. Although this problem remains unsolved, some general patterns have emerged. For each group G there should be a finite set of 'building blocks' UnippGq Ă Irr u pGq called unipotent representations with an array of distinguishing properties. Every representation B P Irr u pGq should be obtained through one of several procedures (like parabolic induction) from a unipotent representation B L P UnippLq of a suitable Levi subgroup L Ă G. Furthermore, the representations in UnippGq should be indexed by nilpotent co-adjoint G-orbits and their equivariant covers. This roadmap has emerged over many decades (see [Vog87, for an overview) and is supported by numerous successes in important special cases (e.g. [Vog86b] and [Bar89]). A crucial problem with this approach is that the set UnippGq has not yet been defined in the appropriate generality. The main contribution of this paper is a definition of 'unipotent' in the case when G is complex. Our definition generalizes the notion of special unipotent, due to Barbasch-Vogan and Arthur ([BV85],[Art83]).The representations we define arise from finite equivariant covers of nilpotent co-adjoint G-orbits. To each such cover r O, we attach a distinguished filtered algebra A 0 equipped with a graded Poisson isomorphism grpA 0 q » Cr rOs. The existence of this algebra follows from the theory of filtered quantizations of conical symplectic singularities, see [Los16]. The algebra A 0 receives a distinguished homomorphism from the universal enveloping algebra U pgq, and the kernel of this homomorphism is a completely prime primitive ideal in U pgq with associated variety O. A unipotent ideal is any ideal in U pgq which arises in this fashion. A unipotent representation is an irreducible Harish-Chandra bimodule which is annihilated (on both sides) by a unipotent ideal.Our definitions are vindicated by the many favorable properties which these ideals and bimodules enjoy. First of all, we show that both unipotent ideals and bimodules have nice geometric classifications. Unipotent ideals are classified by certain geometrically-defined equivalence classes of covers of nilpotent orbits. Unipotent bimodules are classified by irreducible representations of certain finite groups. For classical groups, we show that all unipotent ideals are maximal and all unipotent bimodules are unitary. In addition, we show that all unipotent bimodules are, as G-representations, of a very special form, proving a conjecture of Vogan ([Vog91]). Finally, we show that all special unipotent bimodules are unipotent. The final assertion is proved using a certain refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality of [BLPW16b]. Along the way, we establish combinatorial algorithms (in classical and exceptional types) for computing the infinitesimal characters of unip...

Let g be a simple classical Lie algebra over C and G be the adjoint group. Consider a nilpotent element e ∈ g, and the adjoint orbit O = Ge. The formal slices to the codimension 2 orbits in the closure O ⊂ g are well-known due to the work of Kraft and Procesi [KP82]. In this paper we prove a similar result for the universal G-equivariant cover O of O. Namely, we describe the codimension 2 singularities for its affinization Spec(C[ O]).

In the monograph [LMM21], 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 [BV85] and form the (conjectural) building blocks of the unitary dual. In [LMM21] 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 [AMLV]) we show that if G is a real form of a simple group of exceptional type, all such representations are unitary.

Note that an extremal weight for G in V G is also an extremal weight for M . Thus, Hom M pV G , V M q ‰ 0, as desired.