Dong Liu, Linsheng Zhu
In this paper, we classify all irreducible weight modules with finite dimensional weight spaces over the $W$-algebra $W(2, 2)$. Meanwhile, all indecomposable modules with one dimensional weight spaces over the $W$-algebra $W(2, 2)$ are also determined.
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The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.
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.
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.
In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra $\widehat{\mathfrak{k}}$ and its contact subalgebra $\mathfrak{k}^+$. Based on these results, we give a classification of simple Harish-Chandra modules for $\widehat{\mathfrak{k}}$ and $\mathfrak{k}^+$.
Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized…
Quantum Harish-Chandra bimodules at roots of unity and affine Hecke category
本文研究量子群在q为奇阶单位根时的Harish-Chandra双模范畴，将其与仿射Soergel双模范畴及非交换Springer解联系起来。
It is obtained that an irreducible weight module with finite weight multiplicities over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a so-called finitely-dense module.
On Harish-Chandra's Isomorphism
本文回顾了哈里-昌拉同构及其在表示论和数学物理中的应用，并宣布了任意根系的非对称位移算子的存在性和唯一性。
We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all deformation parameters non-zero q, t, such that t does not equal to +/-i, and q is not a non-trivial root of unity. We also discuss its extension to the root of unity case.
We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi, Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decomposition…
Characteristic cycles,leading term cycles,associated varieties and Harish-Chandra cells are computed for the family of highest weight Harish-Chandra modules for Sp(2n, R) having regular integral infinitesimal character.
Quantum Harish-Chandra isomorphism for the double affine Hecke algebra of $GL_n$
本文证明了在通用参数下，双仿射Hecke代数与量化乘积quiver变种之间存在量子Harish-Chandra同构。
Classification of simple Harish-Chandra modules over the generalized Witt algebras
本文研究并完成了单广义Witt代数上单Harish-Chandra模的分类问题。
In this paper we study a quantum analogue of a degenerate principal series of $U_q \mathfrak{su}_{n,n}$-modules ($0<q<1$) related to the Shilov boundary of the quantum $n \times n$-matrix unit ball. We give necessary and sufficient conditions for the modules to be simple and unitarizable and investigate their equivalence. These results are q-analogues of known classical results on reducibility and unitarizability of SU(n,n)-modules obtained by Johnson, Sahi, Zhang, Howe and Tan.
Harish-Chandra Theorem for the Multi-Parameter Quantum Groups of Okado-Yamane Type
本文研究多参数Okado-Yamane型量子群U_{q,G}(g)的中心，证明Harish-Chandra映射是同构，确定中心Z(U_{q,G})同构于多项式代数或其商代数。
We study Harish-Chandra representations of Yangian for gl(2). We prove an analogue of Kostant theorem showing that resterited Yangians for gl(2) are free modules over certain maximal commutative subalgebras. We also study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.
In this paper, it is proved that all irreducible Harish-Chandra modules over the $\Q$ Heisenberg-Virasoro algebra are of intermediate series (all weight spaces are 1-dimensional).
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.
The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.
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 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.
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.
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.
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
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.
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。
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.
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}
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.
Simple Harish-Chandra bimodules in positive characteristics
本文在正特征代数闭域上，对标准约化群$G$，分类了简单Harish-Chandra双模并确定了其中心特征。
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.
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.
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.
In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.
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.
Harish-Chandra isomorphism for cyclotomic double affine Hecke algebras
本文证实了Braverman–Etingof–Finkelberg猜想，证明分圆双仿射Hecke代数的球面子代数同构于循环箭图的量子化乘法箭图簇，并给出其与平移量子环面代数的同构关系。
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。
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…
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。
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.
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。
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…
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.
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。
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.
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。
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
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 …
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.
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。
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。
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.
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…
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.
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).
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.
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-…
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。
There are several serious errors in this paper. I thought to have time to fix it, but it did not occur for years. So I withdraw this paper.
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.
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