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Register a new userKernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
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A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.
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We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the classical special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on authors habilitation thesis [Spi7] containing a more detailed account of the subject.
Explicit determinant formulas are presented for the $tau$ functions of the generalized Painleve equations of type $A$. This result allows an interpretation of the $tau$-functions as the Plucker coordinates of the universal Grassmann manifold.
In this paper, we explore a canonical connection between the algebra of $q$-difference operators $widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $mathcal{A}$ of the Lie algebra $mathfrak{gl}_{infty}$. We also introduce and study a category $mathcal{O}$ of $widetilde{V}_{q}$-modules. More precisely, we obtain a realization of $widetilde{V}_{q}$ as a covariant algebra of the affine Lie algebra $widehat{mathcal{A}^{*}}$, where $mathcal{A}^{*}$ is a 1-dimensional central extension of $mathcal{A}$. We prove that restricted $widetilde{V_{q}}$-modules of level $ell_{12}$ correspond to $mathbb{Z}$-equivariant $phi$-coordinated quasi-modules for the vertex algebra $V_{widetilde{mathcal{A}}}(ell_{12},0)$, where $widetilde{mathcal{A}}$ is a generalized affine Lie algebra of $mathcal{A}$. In the end, we show that objects in the category $mathcal{O}$ are restricted $widetilde{V_{q}}$-modules, and we classify simple modules in the category $mathcal{O}$.
We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product.
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