Katsuhisa Mimachi

Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials

We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.

A REPRODUCING KERNEL FOR NONSYMMETRIC MACDONALD POLYNOMIALS

We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.

A duality of MacDonald-Koornwinder polynomials and its application to integral representations

We give a formula representing a duality of Macdonald-Koornwinder polynomials. Using this formula, an integral representation of Macdonald-Koornwinder polynomials is derived, a special case of which is the conjectural formula stated in [22]. We also present the corresponding formula to Heckman and Opdam’s Jacobi polynomials of type BCm.

A New Derivation of the Inner Product Formula for the Macdonald Symmetric Polynomials

We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type An-1. As a special case, the corresponding constant term conjecture is also proved.

Barnes-Type Integral and the Meixner–Pollaczek Polynomials

An integral representation of the Meixner–Pollaczek polynomials is presented in terms of a multidimensional generalization of the Barnes type integral. Motivation consists in the study of the Barnes type integrals from the viewpoint of a finite-difference version of the de Rham theory.

A Solution of the Quantum Knizhnik Zamolodchikov Equation of Type Cn

Abstract:We construct a solution of Cheredniks quantum Knizhnik Zamolodchikov equation associated with the root system of type Cn. This solution is given in terms of a restriction of a q-Jordan–Pochhammer integral. As its application, we give an explicit expression of a special case of the Macdonald polynomial of the Cn type. Finally we explain the connection with the representation of the Hecke algebra.