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Zhanqiang Bai

Publications and source records attributed to Zhanqiang Bai.

At least 19 recordsLinked to original sources
arXiv2026

A new proof for the partition algorithm of the annihilator varieties of highest weight modules

Let $L(λ)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $λ-ρ$, where $ρ$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(λ)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(λ)$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.

math.RT↗
arXiv2026

Symmetry Regularization of 1D Generalized Coulomb Problems

For the 1D generalized Coulomb problems--a family that includes the quantizations of the 1D generalized Kepler problems of Ma-Meng-Xiao \cite{MMX2025}--we construct two explicit unitary intertwiners $\hatι_{\pm}$, the quantum analogs of the classical symmetry regularization maps $ι_{\pm}$ of \cite{MMX2025}, that unitarily identify each $H_κ$-energy-definite portion of the Hilbert space $L^{2}(\mathbb{R}_{>0},\mathrm{d}q)$ with a unitary lowest-weight representation of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$.

math-ph↗
arXiv2026

Associated varieties of integral minimal highest weight modules

Let $\mathfrak{g}$ be a complex simple Lie algebra and $L(λ)$ be a highest weight module of $\mathfrak{g}$ with highest weight $λ-ρ$, where $ρ$ is half the sum of positive roots. A simple $\mathfrak{g}$-module $L_w:=L(-wρ)$ is called integral minimal if the associated variety of its annihilator ideal equals the closure of the minimal special nilpotent orbit. In this paper, we find that the associated variety of any integral minimal module $L_w$ is irreducible and equal to the orbital variety corresponding to the minimal length element in the Kazhdan--Lusztig right cell containing $w$.

math.RT↗
arXiv2025

Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties

Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(λ)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $λ$. In this article, we exhibit a bijection between the set of connected Dynkin subdiagrams containing the noncompact simple root and the set of unitary highest weight modules $L(-wρ-ρ)$, where $ρ$ is half the sum of positive roots. We find that $L(-wρ-ρ)$ is unitary if and only if the Schubert variety $X(w)$ is smooth. We also give the cardinality of the set of unitary highest weight modules $L(-wρ-ρ)$ for each Kazhdan-Lusztig right cell.

math.RT↗
arXiv2025

A characterization of Kazhdan-Lusztig right cells containing smooth elements

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$. Its Weyl group is the symmetric group $S_n$. In this paper, we want to describe some Kazhdan-Lusztig right cells containing smooth elements which parameterize the smooth Schubert varieties. These elements are closely related to the study of associated varieties of highest weight modules of $\mathfrak{sl}(n,\mathbb{C})$. Firstly, we give a complete classification of the KL right cells containing only smooth elements. Then we give a sufficient condition for a KL right cell to contain only non-smooth elements by using invariant subsequences and a sufficient condition for a KL right cell to contain some smooth elements. Finally, we give an efficient algorithm to find out all the smooth elements in a given KL right cell.

math.RT↗
arXiv2025

On the cells and associated varieties of highest weight Harish-Chandra modules

Let $G$ be a Hermitian type Lie group with the complexified Lie algebra $\mathfrak{g}$. We use $L(λ)$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $λ$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-wρ-ρ$. In this paper we prove that there is only one Kazhdan--Lusztig right cell such that the corresponding highest weight Harish-Chandra modules $L_w$ have the same associated variety. Then we give a characterization for those $w$ such that $L_w$ is a highest weight Harish-Chandra module and the associated variety of $L(λ)$ will be characterized by the information of the Kazhdan--Lusztig right cell containing some special $w_λ$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.

math.RT↗
arXiv2024

A characterization of socular highest weight modules and Richardson orbits of classical types

Let $\mathfrak{g}$ be a simple complex Lie algebra of classical type with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules are just those highest weight modules with largest possible Gelfand-Kirillov dimension in the corresponding parabolic category $\mathcal{O}^{\mathfrak{p}}$. In this article, we will give an explicit characterization for these modules. When the module is integral, our characterization is given by the information of the corresponding Young tableau associated to the given highest weight module. When the module is nonintegral, we still have some characterization by using the results in the integral case. In our characterization, we define a particular Young diagram called Z-diagram. From this diagram, we can describe the partition type of the unique Richardson orbit associated to the given parabolic subalgebra $\mathfrak{p}$.

math.RT↗
arXiv2024

Associated varieties of minimal highest weight modules

Let $\mathfrak{g}$ be a complex simple Lie algebra. A simple $\mathfrak{g}$-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for $\\mathfrak{g}$. This classification extends the work of Joseph, which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties.

math.RT↗
arXiv2024

On the reducibility of scalar generalized Verma modules associated to two-step nilpotent parabolic subalgebras

Let $\mathfrak{g}$ be a simple complex Lie algebra.A generalized Verma module induced from a one-dimensional representation of a parabolic subalgebra of $\mathfrak{g}$ is called a scalar generalized Verma module of $\mathfrak{g}$. In this article, we use Gelfand-Kirillov dimension to determine the reducibility of scalar generalized Verma modules of $\mathfrak{g}$ associated to a two-step nilpotent parabolic subalgebra of non-maximal type. Such a module exists only when $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(2n,\mathbb{C})$ or $E_6$. We find that the reducible points of these modules can be drawn in a two-dimensional complex plane.

math.RT↗
arXiv2024

A characterization of unitarity of some highest weight Harish-Chandra modules

Let $L(λ)$ be a highest weight Harish-Chandra module with highest weight $λ$. When the associated variety of $L(λ)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(λ)$ can be determined by a simple condition on the value of $z = (λ+ ρ, β^{\vee})$, where $ρ$ is half the sum of positive roots and $β$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules.

math.RT↗
arXiv2024

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

Let $G$ be a Hermitian type Lie group with maximal compact subgroup $K$. Let $L(λ)$ be a highest weight Harish-Chandra module of $G$ with the infinitesimal character $λ$. By using some combinatorial algorithm, we obtain a description of the annihilator variety of $L(λ)$. As an application, when $L(λ)$ is unitarizable, we prove that the Gelfand-Kirillov dimension of $L(λ)$ only depends on the value of $z=(λ,β^{\vee})$, where $β$ is the highest root.

math.RT↗
arXiv2024

An explicit characterization of socular simple modules of $\mathfrak{sl}(n,\mathbb{C})$

Let $\mathfrak{g}$ be a simple complex Lie algebra with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules play an important role in the parabolic versions of category $\mathcal{O}^{\mathfrak{p}}$. From Irving's work, we know that these modules are just those modules with largest possible Gelfand-Kirillov dimension in $\mathcal{O}^{\mathfrak{p}}$. In this article, we will give an explicit characterization for these modules of $\mathfrak{sl}(n,\mathbb{C})$. Our characterization is given in the information of the corresponding highest weight and Young tableau.

math.RT↗
arXiv2024

On the annihilator variety of a highest weight module for classical Lie algebras

Let $\mathfrak{g}$ be a classical complex simple Lie algebra. Let $L(λ)$ be a highest weight module of $\mathfrak{g}$ with highest weight $λ-ρ$, where $ρ$ is half the sum of positive roots. The associated variety of the annihilator ideal of $L(λ)$ is called the annihilator variety of $L(λ)$.It is known that the annihilator variety of any highest weight module $L(λ)$ is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. But in general, this nilpotent orbit is not easy to describe for a given highest weight module $L(λ)$. In this paper, we will give some simple formulas to characterize this unique nilpotent orbit appearing in the annihilator variety of a highest weight module for classical Lie algebras. Our formulas are given by introducing two algorithms, i.e., bipartition algorithm and partition algorithm. To get a special or metaplectic special partition from a domino type partition, we define the H-algorithm based on the Robinson-Schensted insertion algorithm. By using this H-algorithm, we can easily determine this nilpotent orbit from the information of $λ$.

math.RT↗
arXiv2024

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

We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character.

math.RT↗
arXiv2023

Irreducible representations of $\textrm{GL}_n(\mathbb{C})$ of minimal Gelfand-Kirillov dimension

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of $G=\textrm{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition matrix between the two bases for the corresponding coherent families.

math.RT↗
arXiv2023

Quantum PT-Phase Diagram in a Non-Hermitian Photonic Structure

Photonic structures have an inherent advantage to realize PT-phase transition through modulating the refractive index or gain-loss. However, quantum PT properties of these photonic systems have not been comprehensively studied yet. Here, in a bi-photonic structure with loss and gain simultaneously existing, we analytically obtained the quantum PT-phase diagram under the steady state condition. To characterize the PT-symmetry or -broken phase, we define an Hermitian exchange operator expressing the exchange between quadrature variables of two modes. If inputting several-photon Fock states into a PT-broken bi-waveguide splitting system, most photons will concentrate in the dominant waveguide with some state distributions. Quantum PT-phase diagram paves the way to the quantum state engineering, quantum interferences, and logic operations in non-Hermitian photonic systems.

physics.optics↗
arXiv2022

Gelfand-Kirillov dimensions and Reducibility of scalar type generalized Verma modules for classical Lie algebras

Let $\mathfrak{g}$ be a classial Lie algebra and $\mathfrak{p}$ be a maximal parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{p}$. Such $M$ is called a scalar type generalized Verma module. Its simple quotient $L$ is a highest weight moudle. In this paper, we will determine the reducibility of such scalar type generalized Verma modules by computing the Gelfand-Kirillov dimension of $L$.

math.RT↗