Yucai Su
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.
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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 $\mathfrak{k}^+$.
In this paper, we provide a uniform method to thoroughly classify all Harish-Chandra modules over some Lie algebras related to the Virasoro algebras. We first classify such modules over the Lie algebra $W(\varrho)[s]$ for $s=0,\frac12$. With this result and method, we can also do such works for some Lie algebras and superconformal algebras related to the Virasoro algebra, including the several kinds of Schrödinger-Virasoro Lie algebras, which are open up to now.
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。
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.
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…
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。
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。
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.
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 analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker-Akhiezer functions.
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。
We apply a new technique based on double affine Hecke algebras to the Harish-Chandra theory of spherical zonal functions. The formulas for the Fourier transforms of the multiplications by the coordinates are obtained as well as a simple proof of the Harish-Chandra inversion theorem using the Opdam transform.
This paper is devoted to homological treatment of Harish-Chandra decomposition for zonal spherical functions of type $A_n$.
We use the Kazhdan-Lusztig tensoring to define affine translation functors, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA, and to sketch a functorial approach to ``affine Harish-Chandra bimodules''.
The paper mainly considers the center of two-parameter quantum groups $U_{r,s}(\mathfrak{so}_{2n+1})$ via an analogue of the Harish-Chandra homomorphism. In the case when $n$ is odd, the Harish-Chandra homomorphism is not injective in general. When $n$ is even, the Harish-Chandra homomorphism is injective and the center of two-parameter quantum groups $U_{r,s}{(\mathfrak{so}_{2n+1})}$ is described, up to isomorphism.
Harish-Chandra images of orthosymplectic Sugawara operators and Casimir elements
研究正交辛李超代数的西格尔 - 杉原向量，通过哈里什 - 钱德拉同态计算其像并扩展到相关算子和元素，还生成了高阶高丁哈密顿量等，附录回顾了一般线性李超代数类似结果。
Generalized Harish-Chandra morphism on Reflection Equation algebras
本文将广义 Harish-Chandra 态射从包络代数 U(gl(N)) 推广到反射方程代数，通过引入与斜可逆 Hecke 对称对应的生成矩阵特征值，并利用该态射构造权系统的量子类比。
We present a simple proof of a strengthening of the derived Beilinson-Bernstein localization theorem using the formalism of descent in derived algebraic geometry. The arguments and results apply to arbitrary modules without the need to fix infinitesimal character. Roughly speaking, we demonstrate that all Ug-modules are the invariants, or equivalently coinvariants, of the action of intertwining functors (a refined form of Weyl group symmetry). This is a quantum version of descent for the Grothendieck-Springer simultaneous resolution. In an appendix we present an alternative perspective, which …
In this paper we prove that certain matrix elements of vertex operators of deformed W-algebra satisfy Macdonald difference equations and form n! -dimensional space of solutions. These solutions are the analogues of Harish Chandra solutions with prescribed asymptotic behavior. We obtain formulas for analytic continuation as a consequence of braiding properties of vertex operators of deformed W-algebra.
On Harish-Chandra modules over quantizations of nilpotent orbits
本文针对复数域上半单代数群的幂零轨道代数过滤量子化上的满支撑不可约Harish-Chandra模，给出几何分类嵌入映射，证明部分情形下为双射并刻画非双射情形的像，还完成例外李代数的部分分类。
Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednik's root system analogs of the quantum Knizhnik-Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang-Baxter and reflection equations. Applications to …
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。
This volume contains a mildly expanded version of lectures and talks at seminars and conferences, as well as review papers on subjects listed in the title of the volume. A great deal of these texts have already been published or sent to press. However, the only way to provide a good exposition of the field we are interested in is to collect all of those papers together. Bounded symmetric domains form a favorite subject of research in function theory, non-commutative harmonic analysis, and representation theory of real reductive Lie groups. The authors introduce the notions of q-analogues of bo…
It is proved that an indecomposable Harish-Chandra module over the Virasoro algebra must be (i) a uniformly bounded module, or (ii) a module in Category $\cal O$, or (iii) a module in Category ${\cal O}^-$, or (iv) a module which contains the trivial module as one of its composition factors.
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