A geometric decomposition of Harish-Chandra modules

Parabolic induction and the Harish-Chandra D-module

Let G be a reductive group and L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L=T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D-module on G x T. We show that this module is flat over D(T), whi…

On modular Soergel bimodules, Harish-Chandra bimodules, and category O

In this paper we continue the study of the category of modular Harish-Chandra bimodules initiated by Bezrukavnikov and Riche and also study the modular version of the BGG category $\mathcal{O}$. We prove a version of the Bezrukavnikov-Mirkovic-Rumynin localization theorem for the Harish-Chandra bimodules and for the category $\mathcal{O}$. We also relate the category of Harish-Chandra bimodules to the affine Hecke category building on the prior work of Bezrukavnikov and Riche.

Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$

In this paper, using the theory of $\A$-cover developed in \cite{B1,BF1}, we completely classify all simple Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$. As a byproduct, we obtain the classification of simple Harish-Chandra modules over the classical $W$-algebra $W(2,2)$ studied in \cite{LGZ,CLW,GLZ1}.

Characteristic cycles of highest weight Harish-Chandra modules

Characteristic cycles and leading term cycles of irreducible highest weight Harish-Chandra modules of regular integral infinitesimal character are determined. In the simply laced cases they are irreducible, but in the nonsimply laced cases they are more complicated.

Examples of leading term cycles of Harish-Chandra modules

Many examples are given of irreducible Harish-Chandra modules for type C having reducible leading term cycles. Examples are also given of irreducibles in category O having reducible associated varieties in type C.

Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules

We show that the coefficients of Kazhdan-Lusztig-Vogan polynomials attached to certain blocks of Harish-Chandra modules satisfy a monotonicity property relative to the closure order on K-orbits in the flag variety.

K-finite matrix elements of irreducible Harish-Chandra modules are hypergeometric

We show that all the K-finite matrix elements of irreducible Harish-Chandra modules can can be expressed in spherical functions using finite number of operations

$SL_2(\mathbb{R})$ 的 Harish-Chandra 对常值族的扩展

Extensions of the constant family of Harish-Chandra pairs of $SL_2(\mathbb{R})$

本文研究并分类了复仿射线和复投影线上、一般纤维同构于 $SL_2(\mathbb{R})$ 的 Harish-Chandra 对的代数族。

On the Howe correspondence and Harish-Chandra series

We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs ($\mathbf{U}_m(\mathbb{F}_q),\mathbf{U}_n(\mathbb{F}_q$)) and ($\mathbf{Sp}_{2m}(\mathbb{F}_q),\mathbf{O}^\pm_{2n}(\mathbb{F}_q)$), where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd). We use the Lusztig correspondence to reduce the study of the Howe correspondence between Harish-Chandra series to a correspondence between unipotent Harish-Chandra series. Finally, we provide two applications of these results.

Harish-Chandra模上的典范滤过

Canonical filtrations on Harish-Chandra modules

本文证明包括所有不可约模在内的一大类Harish-Chandra (g,K)-模都具有某种典范滤过结构。

Annihilators and associated varieties of Harish-Chandra modules for Sp(p,q)

We show how to compute annihilators and associated varieties of simple Harish-Chandra modules for Sp(p,q) with trivial infinitesimal character from a pair of domino tableaux attached to a parameter for such a module.

Classification of simple Harish-Chandra modules over the N=1 Ramond algebra

In this paper, we give a new approach to classify all simple Harish-Chandra modules for the N=1 Ramond algebra based on the so called A-cover theory developed in \cite{BF}

Smooth Frechet Globalizations of Harish-Chandra Modules

An elementary and self-contained approach to smooth Frechet globalizations of Harish-Chandra modules is presented. We provide applications to the theory of Eisenstein series and Automatic Continuity.

Multi-fusion categories of Harish-Chandra bimodules

We survey some results on tensor products of irreducible Harish-Chandra bimodules. It turns out that such tensor products are semisimple in suitable Serre quotient categories. We explain how to identify the resulting semisimple tensor categories and describe some applications to representation theory.

Harish-Chandra series in finite unitary groups and crystal graphs

The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

Harish-Chandra bimodules in the Deligne category $\text{Rep}(GL_t)$

In this paper we study the category of Harish-Chandra bimodules $HC_{χ,ψ}$ in the Deligne category $\text{Rep}(GL_t)$. In particular, we answer Question 3.25 posed in Pavel Etingof's paper arXiv:1407.0373 and determine for which central characters $χ$ and $ψ$ this category is not zero.

Harish-Chandra highest weight representations of semisimple Lie algebras and Lie groups

In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.

A Paley-Wiener theorem for Harish-Chandra modules

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.

关于哈里-查兰德拉某些结果的表示理论研究，扩展到其中心扩张

On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions

本文完成哈里-查兰德拉关于p-进群的parabolic诱导不可约性与mu函数解析行为的联系证明，并展示该证明在中心扩张情况下的有效性。

Spherical principal series of quantum Harish-Chandra modules

The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.

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…

关于最高权 Harish-Chandra 模的结合簇

On the associated variety of a highest weight Harish-Chandra module

本文证明了直接从最高权计算最高权 Harish-Chandra 模结合簇的公式，并给出了适用于任意无穷小特征且在所有 Cartan 类型中统一的 Gelfand--Kirillov 维数公式。

On the existence of infinite-dimensional generalized harish-chandra modules

We prove a general existence result for infinite-dimensional admissible (g;k)-modules, where g is a reductive finite-dimensional complex Lie algebra and k is a reductive in g algebraic subalgebra.

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。

Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularity

We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for ${\mathrm{GL}}(8,\mathbb{C})$ with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for ${\mathfrak{g}}{\mathfrak{l}}(12,\mathbb{C})$ with reducible associated varieties that Williamson discovered.

Generalized Harish-Chandra modules with generic minimal $\frak k$-type

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras $(\g,\k)$, we construct, via cohomological induction, the fundamental series $F^\cdot (\p,E)$ of generalized Harish-Chandra modules. We then use $F^\cdot (\p,E)$ to characterize any simple generalized Harish-Chandra module with generic minimal $\k$-type. More precisely, we prove that any such simple $(\g,\k)$-module of finite type arises as the unique simple submodule o…

Harish-Chandra pairs in the Verlinde category in positive characteristic

In this article, we prove that the category of affine group schemes of finite type in the Verlinde category is equivalent to the category of Harish-Chandra pairs in the Verlinde category. Subsequently, we extend this equivalence to an equivalence between corresponding representation categories and then study some consequences of this equivalence to the representation theory of $GL(L)$, with $L$ a simple object in the Verlinde category.

Algebraic Characters for Harish-Chandra modules

We give a cohomological treatment of a character theory for (g,K)-modules. This leads to a nice formalism extending to large categories of not necessarily admissible (g,K)-modules. Due to results of Hecht, Schmid and Vogan the classical results of Harish-Chandra's global character theory extend to this general setting. As an application we consider a general setup, for which we show that algebraic characters answer discretely decomposable branching problems.

关于联系 Harish-Chandra 双模与 Soergel 双模的函子

On the functor relating Harish-Chandra bimodules and Soergel bimodules

本文重新考察 Soergel 函子，将其与 Losev 的限制函子联系起来，计算相关双模的像并给出函子性质的新证明，同时把整块结果推广到一般块。

Algebraic supergroups and Harish-Chandra pairs over a commutative ring

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi and Gavarini [8] and by Gavarini [9, 10]. For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras.

超对称空间与幽灵分布的Harish-Chandra同构

The Harish-Chandra isomorphism for supersymmetric spaces and ghost distributions

本文证明了超对称空间的Harish-Chandra同构定理，描述了不变微分算子特征值的多项式代数，并证明了满足“平方根”不变性条件的幽灵分布的Harish-Chandra同构。

Unitary Harish-Chandra representations of real supergroups

We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.

On modular Harish-Chandra series of finite unitary groups

In the modular representation theory of finite unitary groups when the characteristic $\ell$ of the ground field is a unitary prime, the $\hat{\mathfrak{sl}}_e$-crystal on level $2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $\ell$ sufficiently large, the $\mathfrak{sl}_\infty$-crystal on the same level $2$ Fock spaces provides the remaining piece of the puzzle for the …

Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field

Over an arbitrary field of characteristic $\ne 2$, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.

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。

映射全环面李代数上不可约Harish-Chandra模的分类

Classification of irreducible Harish-Chandra modules over map full toroidal Lie algebras

本文分类了映射全环面李代数上的不可约Harish-Chandra模，证明其必为尖点模或最高权模，且均为单点赋值模。

Towards a tensor-classification of Harish-Chandra modules: The case ${\rm SL}_2(\mathbb R)$

We consider the category of Harish-Chandra modules for ${\rm SL}_2(\mathbb R)$ as a module over the category of finite-dimensional representations of ${\rm SL}(2)$ with respect to the tensor product. In this note we use classical results about principal series to obtain a classification of $\otimes$-submodules of the category of Harish-Chandra modules of ${\rm SL}_2(\mathbb R)$. The resulting classification recovers the classical classification of irreducible Harish-Chandra modules. Our methods are expected to generalize to arbitrary reductive pairs and more general base fields.

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…

Harish-Chandra Cuspidal Pairs

The irreducible characters of a finite reductive group are partitioned into Harish-Chandra series that are labelled by cuspidal pairs. In this note, we describe how one can algorithmically calculate those cuspidal pairs using results of Lusztig.

A generalized Harish-Chandra isomorphism

For any complex reductive Lie algebra g and any locally finite g-module V, we extend to the tensor product of U(g) with V the Harish-Chandra description of g-invariants in the universal enveloping algebra U(g).

Deformed Harish-Chandra homomorphism for the cyclic quiver

In the case of cyclic quiver we prove that the deformed Harish-Chandra map whose existence was conjectured by Etingof and Ginzburg is well defined. As an application we prove Kirillov-type formula for the cyclotomic Bessel function.

On the Harish-Chandra Homomorphism for Quantum Superalgebras

In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra $\mathrm{U}_q(\mathfrak{g})$ associated with a simple basic Lie superalgebra $\mathfrak{g}$ and give an explicit description of its image. We use it to prove that the center of $\mathrm{U}_q(\mathfrak{g})$ is isomorphic to a subring of the ring $J(\mathfrak{g})$ of exponential super-invariants in the sense of Sergeev and Veselov, establishing a Harish-Chandra type theorem for $\mathrm{U}_q(\mathfrak{g})$. As a byproduct, we obtain a basis of the center of $\mathrm{U}_q(\mathfrak{g})$ with the aid of quasi-…

论Harish-Chandra块模的范畴

On the Category of Harish-Chandra Block Modules

该研究去掉Harish-Chandra子代数的拟交换假设，引入等价关系定义Harish-Chandra块模与块子代数，给出块模范畴分解及与有限A-模的范畴等价，推广并改进了前人结果，还给出单块模同构类有限的充分条件。

Harish-Chandra bimodules for quantized Slodowy slices

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizaions of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, and to construct `noncommutative resolutions' of Slodowy slices via translation functors.

Annihilators and associated varieties of Harish-Chandra modules for $SO^*(2n)$

Given a simple Harish-Chandra module for this group of trivial infinitesimal character, parametrized by a signed involution, we produce combinatorial recipes for attaching to it a pair of tableaux from which its annihilator and associated variety can be read off.

哈里-钱德拉诱导与特征标Jordan分解

Harish-Chandra Induction and Jordan Decomposition of Characters

研究证明有限连通可约群中总能选择Jordan分解与哈里-钱德拉诱导相容，并推广了中心连通群的经典结果。

Generalized Harish-Chandra Modules: A New Direction

Let $\frak g$ be a reductive Lie algebra over $\bold C$. We say that a $\frak g$-module $M$ is a generalized Harish-Chandra module if, for some subalgebra $\frak k \subset\frak g$, $M$ is locally $\frak k$-finite and has finite $\frak k$-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when $\frak k$ is a Cartan subalgebra. We also review the recent determination of which reductive in $\frak g$ subalgebras $\frak k$ are essential to a classification. Finall…

Classification of simple strong Harish-Chandra modules over the Lie superalgebra of vector fields on $\C^{m|n}$

In this paper, we classify simple strong Harish-Chandra modules over the Lie superalgebra $W_{m,n}$ of vector fields on $\C^{m|n}$. Any such module is the unique simple submodule of some tensor module $F(P,V)$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$ and a simple weight module $V$ over the general linear superalgebra $\gl_{m,n}$.

Harish-Chandra Modules Over the Twisted Heisenberg-Virasoro Algebra

In this paper, we classify all indecomposable Harish-Chandra modules of the intermediate series over the twisted Heisenberg-Virasoro algebra. Meanwhile, some bosonic modules are also studied.

Simple Harish-Chandra modules over the super affine-Virasoro algebras

In this paper, we classify all simple Harish-Chandra modules over the super affine-Virasoro algebra $\widehat{\mathcal{L}}=\mathcal{W}\ltimes(\mathfrak{g}\otimes \mathcal{A})\oplus \mathbb{C}C$, where $\mathcal{A}=\mathbb{C}[t^{\pm 1}]\otimesΛ(1)$ is the tensor superalgebra of the Laurent polynomial algebra in even variable $t$ and the Grassmann algebra in odd variable $ξ$, $\mathcal{W}$ is the Lie superalgebra of superderivations of $\mathcal{A}$, and $\mathfrak{g}$ is a finite-dimensional perfect Lie superalgebra.

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。

On Harish-Chandra theory of global nonsymmetric functions

This paper is devoted to the Harish-Chandra-type decomposition of the global nonsymmetric spherical functions in terms of their asymptotic expansions and the q,t-generalization of the celebrated c-function. This is for any reduced root systems in the q,t-setting; we pay special attention to the case of A1, where this decomposition is very explicit.

最高权Harish-Chandra模的Dirac不等式 II

Dirac inequality for highest weight Harish-Chandra modules II

本文针对Hermitian型非紧致例外单李群，研究用于分类酉最高权模的Dirac不等式。

关于Riemannian对称空间上的Harish-Chandra的Plancherel定理

On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces

本文介绍了Riemannian对称空间上Plancherel理论的方法及Harish-Chandra定理的证明。