相关论文: The Capelli problem for $\mathfrak{gl}(m|n)$ and t…

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 establish a duality of the non-periodic Gaudin model associated with superalgebra $\mathfrak{gl}_{m|n}$ and the non-periodic Gaudin model associated with algebra $\mathfrak{gl}_k$. The Hamiltonians of the Gaudin models are given by…

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…

We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of…

A factorial analogue of the supersymmetric Schur functions is introduced. It is shown that factorial versions of the Jacobi--Trudi and Sergeev--Pragacz formulae hold. The results are applied to construct a linear basis in the center of the…

We consider a see-saw pair consisting of a Hermitian symmetric pair (G_R, K_R) and a compact symmetric pair (M_R, H_R), where (G_R, H_R) and (K_R, M_R) form real reductive dual pairs in a large symplectic group. In this setting, we get…

Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a…

We introduce and study quantum Capelli operators inside newly constructed quantum Weyl algebras associated to three families of symmetric pairs. Both the center of a particular quantized enveloping algebra and the Capelli operators act…

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"…

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli…

To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called $4$-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In…

We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The…

In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier…

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 extend the Capelli identities (1890) from the Lie algebra $gl_N$ to the other two classical Lie algebras $so_N$ and $sp_N$. We employ the theory of reductive dual pairs due to Howe. Our technique comes from the representation theory of…

In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA),…

We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g. gl(m|n)) over the complex numbers. Using results from invariant theory, we show that there exist subsuperalgebras which detect the cohomology of $\mathfrak{g}.$…

A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on…

A depending on a complex parameter $k$ superanalog ${\mathcal S}{\mathcal L}$ of Calogero operator is constructed; it is related with the root system of the Lie superalgebra ${\mathfrak{gl}}(n|m)$. For $m=0$ we obtain the usual Calogero…

Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product,…