Related papers: The Dunkl-Laplace transform and Macdonald's hyperg…

Dunkl operators are differential-difference operators on $\b R^N$ which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we…

We analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of…

This article is devoted to the noncommutative version of the Laplace transformation. New types of the direct and inverse transformations of the Laplace type over general Cayley-Dickson algebras, in particular, also the skew field of…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

We construct and study an explicit simultaneous $\mathscr{Y}$-eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a…

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the…

Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations…

Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an…

We use a new method to study the Laplace-Beltrami type operator on the Fock space of symmetric functions, and as an example of our explicit computation we show that the Jack symmetric functions are the only family of eigenvectors of the…

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial…

Heckman-Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since…

We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the…

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups,…

Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group…

We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation…

We extend some classical results - such as Quillen's Theorem A, the Grothendieck construction, Thomason's Theorem and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads…

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by…

In this article, we define the Fourier-Dunkl transform, which generalizes the Fourier transform. We prove Strichartz's restriction theorem for the Fourier-Dunkl transform for a cone-hyper-surface and its generalisation to the family of…

We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two quasi-periods. They are natural analogues of the…

A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$.…