Related papers: Spherical Functions of Fundamental $K$-Types Assoc…

We construct a generalization of the theory of symmetric functions involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal…

We explicitly establish a unitary correspondence between spherical irreducible tensor operators and cartesian tensor operators of any rank. That unitary relation is implemented by means of a basis of integer-spin wave functions that…

We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of…

By using the $K$-free complex bosons and the $K$-free complex fermions, we construct the ${\cal N}=2$ supersymmetric $W_{\infty}^{K,K}$ algebra which is the matrix generalization of previous ${\cal N}=2$ supersymmetric $W_{\infty}$ algebra.…

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m),…

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape invariant operators. These operators can…

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall-Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to P and Q…

We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…

We derive a sharp inequality relating the second and fourth elementary symmetric functions of the eigenvalues of a trace-free matrix and give two applications. First, we give a new proof of the classification of conformally flat…

Y. Hironaka introduced the spherical functions on the p-adic space of Hermitian matrices. For the space of 2\times2 Hermitian matrices, we complete Hironaka's work by also considering the case of a wildly ramified quadratic extension. We…

We relate two fundamental enumerative functions, namely the $I$-functions in the quantum $K$-ring of $G(r,n)$ and of its cotangent bundle, by defining a $K$-theoretic operator on classes, called balancing. This operator lifts the…

We introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving $\mathbb{Z}_2$-graded reflection equation.…

The $K$-theoretic Schur $P$- and $Q$-functions $GP_\lambda$ and $GQ_\lambda$ may be concretely defined as weight generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of…

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The…

The discovery of integrable $N=2$ supersymmetric Landau-Ginzburg theories whose chiral rings are fusion rings suggests a close connection between fusion rings, the related Landau-Ginzburg superpotentials, and $N=2$ quantum integrability. We…

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion…

This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated…

Invariant integrals of functions and forms over $q$ - deformed Euclidean space and spheres in $N$ dimensions are defined and shown to be positive definite, compatible with the star - structure and to satisfy a cyclic property involving the…

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…