Related papers: Restrictions of generalized Verma modules to symme…

We give branching formulas from $so(7,\mathbb{C})$ to $\mathfrak{g}_2$ for generalized Verma modules attached to $\mathfrak{g}_2$-compatible parabolic subalgebras of $so(7,\mathbb{C})$, and branching formulas from $\mathfrak{g}_2$ to…

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…

We discuss the branching problem for generalized Verma modules $M_\lambda(\mathfrak g, \mathfrak p)$ applied to couples of reductive Lie algebras $\bar{\mathfrak g}\hookrightarrow \mathfrak g$. The analysis is based on projecting character…

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions…

We consider a general parabolic category $\mathcal{O}_{S}$ over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on…

Let $\mathfrak{g}$ be a classical complex simple Lie algebra and $\mathfrak{q}$ be a parabolic subalgebra. Generalized Verma module $M$ is called a scalar generalized Verma module if it is induced from a one-dimensional representation of…

The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is…

A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the…

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…

Let g be a exceptional complex simple Lie algebra and q be a parabolic subalgebra. A generalized Verma module M is called a scalar generalized Verma module if it is induced from a one-dimensional representation of q. In this paper, we will…

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the…

The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in \cite{ms}. In the present article, we employ the recently developed F-method,…

Let $\mathfrak g(G,\lambda)$ denote the deformed generalized Heisenberg-Virasoro algebra related to a complex parameter $\lambda\neq-1$ and an additive subgroup $G$ of $\mathbb C$. For a total order on $G$ that is compatible with addition,…

In this article, we study the restriction of Zuckerman's derived functor (g,K)-modules A_q(\lambda) to g' for symmetric pairs of reductive Lie algebras (g,g'). When the restriction decomposes into irreducible (g',K')-modules, we give an…

Let $g$ be a semisimple Lie algebra over $\mathbb C$ and $k$ be a reductive in $g$ subalgebra. We say that a simple $g$-module $M$ is a $(g; k)$-module if as a $k$-module $M$ is a direct sum of finite-dimensional $k$-modules. We say that a…

Let $G$ be a noncompact connected simple Lie group, and $(G,G^\Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for…

Let G be a semi simple linear algebraic group over a field of characteristic zero and let V be a finite dimensional irreducible G-module with highest weight vector v. Let P in G be the parabolic subgroup fixing v and let g=Lie(G). We get a…

In this paper, we show that the Dirac cohomology $H_{D}(L(\lambda))$ of a simple highest weight module $L(\lambda)$ in $\mathcal{O}^\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\mathcal{W}_I(\lambda)$ of the…

Let $K$ be an algebraically closed field of characteristic zero, and let $G$ be a connected reductive algebraic group over $K$. We address the problem of classifying triples $(G,H,V)$, where $H$ is a proper connected subgroup of $G$, and…

Let $(G,G^\Gamma)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^\Gamma)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $\pi_K$ that is…