Harish-Chandra bimodules for quantized Slodowy slices

On unitarizable Harish-Chandra bimodules for deformations of Kleinian singularities

The notion of a Harish-Chandra bimodule, i.e. finitely generated $U(\mathfrak{g})$-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [Ivan Losev, Dimensions of irreducible modules over W-algebras and Goldie ranks. arXiv:1209.1083]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, i.e. the algebra itself, for a deformation of Kleinian singularity of type $A$ is unitarizable. We obtain a partial classification of unitarizable regular bimodules.

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the cate…

Harish-Chandra bimodules of finite $K$-type in Deligne categories

We continue the study of Harish-Chandra bimodules in the setting of the Deligne categories $\mathrm{Rep}(G_t)$, that was started in the previous work of the first author (arXiv:2002.01555). In this work we construct a family of Harish-Chandra bimodules that generalize simple finite dimensional bimodules in the classical case. It turns out that they have finite $K$-type, which is a non-vacuous condition for the Harish-Chandra bimodules in $\mathrm{Rep}(G_t)$. The full classification of (simple) finite $K$-type bimodules is yet unknown. This construction also yields some examples of central char…

具有极小支撑的量子化花箭簇簇上的Harish-Chandra双模

Harish-Chandra bimodules over quantized flower quiver varieties with minimal support

本研究对具有极小支撑的量子化花箭簇簇上的Harish-Chandra双模进行分类，证明维数向量为$n$时，仅当整数量化参数足够大或足够小时存在$n!$个极小支撑单Harish-Chandra双模，核心工具为增强版限制函子。

有理Cherednik代数上的Harish-Chandra双模

Harish-Chandra bimodules over rational Cherednik algebras

本文利用局部化、Namikawa-Weyl群作用和部分KZ函子，研究有理Cherednik代数上的Harish-Chandra双模，分类满支集双模存在的参数对，并在对称群情形分类所有不可约双模。

A型有理Cherednik代数的Harish-Chandra双模

Harish-Chandra bimodules for type A rational Cherednik algebras

本文研究A型有理Cherednik代数的Harish-Chandra双模，证明其范畴可完全忠实嵌入范畴O_c，并在Nakajima箭图簇量子化的一般框架下构造对偶，进而描述含正则双模的最小Serre子范畴及其不可约对象的张量积。

Harish-Chandra bimodules over quantized symplectic singularities

In this paper we classify the irreducible Harish-Chandra bimodules with full support over filtered quantizations of conical symplectic singularities under the condition that none of the slices to codimension 2 symplectic leaves has type $E_8$. More precisely, we show that the top quotient $\overline{\operatorname{HC}}(\mathcal{A}_λ)$ of the category of Harish-Chandra bimodules over the quantization $\mathcal{A}_λ$ with parameter $λ$ embeds into the category of representations of the algebraic fundamental group, $Γ$, of the open leaf. The image coincides with the representations of $Γ/Γ_λ$, whe…

Character D-modules via Drinfeld center of Harish-Chandra bimodules

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg. Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution …

广义Weyl和$q$-Weyl代数上Harish-Chandra双模的单位化性

Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras

本文对广义Weyl和$q$-Weyl代数上的Harish-Chandra双模，在一般量子化参数下分类了不变正定Hermite形式，从而解决其单位化性问题。

A categorical approach to dynamical quantum groups

We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it…

A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gl(n). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed. We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical …

单侧理想与哈里-查恩双模

Unipotent Ideals and Harish-Chandra Bimodules

本文定义了单侧表示，通过nilpotent共轭轨道的有限等变覆盖，结合滤代数和泊松同构，证明了单侧理想和表示的性质，包括参数化和单位性。

Finite dimensional representations of W-algebras

W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish-Chandra bimodules and bimodules over $W$-algebras.

Regular strongly typical blocks of $\mathcal{O}^{\mathfrak{q}}$

We use the technique of Harish-Chandra bimodules to prove that regular strongly typical blocks of the category $\mathcal{O}$ for the queer Lie superalgebra $\mathfrak{q}_n$ are equivalent to the corresponding blocks of the category $\mathcal{O}$ for the Lie algebra $\mathfrak{gl}_n$.

A 型逆哈密顿约化与仿射格拉斯曼流形中的广义切片

Inverse Hamiltonian reduction in type A and generalized slices in the affine Grassmannian

本文通过仿射格拉斯曼流形中广义切片的几何方法，结合 Fedosov 量子化，证明了 A 型有限 W-代数的逆哈密顿约化，并将其推广到 Harish Chandra 双模的 Kostant-Whittaker 约化。

A型量化Kleinian奇点上的扭曲迹与正定形式

Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A

本文研究了A_{n-1}型Kleinian奇点量子化代数上的扭曲迹，给出了显式积分公式以判定正定埃尔米特形式，从而分类了幺正短星积，证实了n≤4时的唯一性猜想，并推广了离散Painlevé系统。

来自 $Φ$-尖点对与仿射 Springer 纤维的层-秩对偶

Level-Rank Dualities from $Φ$-Cuspidal Pairs and Affine Springer Fibers

本文利用 $Φ$-尖点对、Hecke 代数与有理 DAHA 提出一般有限约化群的层-秩对偶猜想，并在 $\mathbb{GL}_n$ 互素情形及多类例外群中加以证明或数值验证。