Related papers: Explicit formulas for Eigenvalues of Capelli opera…

Let $\mathfrak g$ be either the Lie superalgebra $\mathfrak{gl}(V)\oplus\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|n}$ or the Lie superalgebra $\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|2n}$. Furthermore, let $W$ be the $\mathfrak g$-module…

We apply the technique of affine Hecke algebras to the invariant theory of the "queer" Lie superalgebra $q_N$. We give explicit formulas for the elements of a distinguished basis in the centre of $U(q_N)$, determined by "vanishing"…

For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak…

Let $(V,\omega)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,\omega)$ denote the Lie superalgebra of similitudes of $(V,\omega)$. When the space $\mathscr P(V)$ of superpolynomials on $V$…

We classify anti-involutions of Lie superalgebra $\hsd$ preserving the principal gradation, where $\hsd$ is the central extension of the Lie superalgebra of differential operators on the super circle $S^{1|1}$. We clarify the relations…

The character formula of any finite dimensional irreducible module $L_\lambda$ for Lie superalgebra $\mathfrak{osp}(n|2)$ is computed. As a by-product, the decomposition of tensor module $L_\lambda\otimes \mathbb{C}^{n|2}$, where…

We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{\lambda,\nu;\mu}$ of bilinear differential operators acting…

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this…

It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A kappa(s) number of polinomials which depend on rank N are obtained explicitly for A_N Casimir operators of…

The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…

A Fock space is introduced that admits an action of a quantum group of type A supplemented with some extra operators. The canonical and dual canonical basis of the Fock space are computed and then used to derive the finite-dimenisonal…

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the…

Based on the particular orderings introduced for the positive roots of finite dimensional basic Lie superalgebras, we construct the explicit differential operator representations of the $osp(2r|2n)$ and $osp(2r+1|2n)$ superalgebras and the…

We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the…

The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack…

For each positive integer $n$, the basic classical complex Lie superalgebra $\mathfrak{osp}(1|2n)$ has a unique equivalence class of infinite-dimensional completely-pointed modules, those weight modules with one-dimensional weight spaces.…

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is…

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible…

We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the…

We analyse the Witten-Woronowicz's type deformations of the Lie superalgebra osp(2,2) and obtain a deformation parametrized by three independent parameters. For some of these algebras, finite dimensional representations are formulated in…