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...
We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev ([12]). We explore the applications of this theory to unipotent representations. We observe that the machinery of microlocalization provides an alternative characterization of unipotent representations. For G R complex (and under a codimension condition on ∂O), we deduce a formula for the K-multiplicities of unipotent representations attached to a nilpotent orbit O, proving an old conjecture of Vogan ([18]) in a large family of cases.
We consider Arthur's conjectures for split reductive p-adic groups from the point of view of the wavefront set, a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with 'real infinitesimal character' and determine a lower bound for this invariant in terms of the Kazhdan-Lusztig parameters. We define certain unipotent Arthur packets consisting of representations with minimal allowable wavefront sets, and we prove that their Iwahori-spherical constituents are the Aubert-Zelevinsky duals of tempered representations.
Let Gpkq be a semisimple p-adic group, inner to split. In this article, we compute the algebraic and canonical unramified wavefront sets of the irreducible supercuspidal representations of Gpkq in Lusztig's category of unipotent representations.
In this paper, we construct and classify the (special) unipotent representations of a real reductive group attached to the principal nilpotent orbit. Our construction gives rise to a formula for the K-types and associated varieties of all such representations.
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