Harish-Chandra Bimodules Over Quantized Symplectic Singularities

Losev, Ivan

doi:10.1007/s00031-020-09638-5

Cited by 7 publications

(10 citation statements)

References 75 publications

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“…This embedding is full and monoidal, and its image in Γ -mod is closed under taking direct summands, see [Los21,Lem 4.6]. Thus, Imp‚ : q " Γ{Γpλq -mod for a normal subgroup Γpλq Ď Γ, which is uniquely determined by λ.…”

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

“…The second main result is a classification of Harish-Chandra bimodules with full support over a filtered quantization of a conical symplectic singularity. In Section 4.13 we recall a category equivalence, first obtained in [Los21], between the category of Harish-Chandra bimodules with full support and the category of representations of a certain finite group. In Sections 4.14, 4.15 we provide a (partial) description of this finite group (see, in particular, Corollary 4.14.4, Theorem 4.15.2).…”

Section: Moreover If Rmentioning

confidence: 99%

“…For any B P HCpA λ q, the kernel and cokernel of the unit homomorphism B Ñ pB : q : are supported on X sing , see [Los21,Lem 4.6]. Hence, ‚ : : Γ -mod Ñ HCpA λ q is a left inverse for ‚ : : HCpA λ q Ñ Γ -mod, and there are quasi-inverse equivalences (4.13.2)…”

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

“…Let X be a conical symplectic singularity and let A λ be a filtered quantization of CrXs. In this section, we will give a description of the normal subgroup Γpλq Ă Γ, following [Los21]. First, suppose X " C 2 {Γ is a Kleinian singularity.…”

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

“…Then XpΓ{Γpλqq Ă XpΓq corresponds to W ae λ {W a λ Ă W ae {W a under the isomorphism (4.15.1). If Γ is not of type E 8 , Proposition 4.15.2 can be used to compute Γpλq, see [Los21,Section 5]. This result is rather technical and will not be used in this paper, so we omit it.…”

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

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Unipotent Ideals and Harish-Chandra Bimodules

Losev¹,

Mason-Brown²,

Matvieievskyi³

2021

Preprint

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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...

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Section: Hamiltonian Quantizationsmentioning

confidence: 99%

Section: Moreover If Rmentioning

confidence: 99%

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

Section: Hamiltonian Quantizationsmentioning

confidence: 99%

See 3 more Smart Citations

Unipotent Ideals and Harish-Chandra Bimodules

Losev¹,

Mason-Brown²,

Matvieievskyi³

2021

Preprint

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One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method

Topley

2023

Invent. math.

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In this paper we study the variety of one dimensional representations of a finite $W$ W -algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the $W$ W -algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction.

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Unipotent ideals for spin and exceptional groups

Mason-Brown

Matvieievskyi

2023

Journal of Algebra

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Harish-Chandra Bimodules Over Quantized Symplectic Singularities

Cited by 7 publications

References 75 publications

Unipotent Ideals and Harish-Chandra Bimodules

Unipotent Ideals and Harish-Chandra Bimodules

One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method

Unipotent ideals for spin and exceptional groups

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