We construct and study some finite dimensional modules for rational Cherednik algebras for the groups G(r, p, n) by using intertwining operators and a commutative family of operators introduced by Dunkl and Opdam. The coinvariant ring and an analog of the ring constructed by Gordon in the course of proving Haiman’s conjectures on diagonal coinvariants are special cases. We study a certain hyper...

The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S of W corresponds to a standard module M(λ) for H. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a co...

In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of ...

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated fini...

In the paper we calculate the images of the operators of multiplication by Laurent polynomials with respect to the Harish-Chandra transform and its non-symmetric generalization due to Opdam. It readily leads to a new simple proof of the Harish-Chandra inversion theorem in the zonal case (see [HC,He1]) and the corresponding theorem from [O1]. We assume that k > 0 and restrict ourselves to compac...

The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group W . Each irreducible representation S of W corresponds to a standard module M(λ) for H. This paper deals with the infinite family G(r, p, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Du...

This paper uses some basic concepts and results on harmonic analysis associated with Heckman-Opdam theory \(\mathbb {R}^{d}\) to study types of wavelet packets the corresponding transforms. More precisely, we two transforms related hypergeometric Fourier transform W-invariant functions, prove Plancherel, Calderón, reconstruction formulae for these

In this talk we describe some applications of Procesi bundles that appeared in Gufang’s talk to type A Rational Cherednik algebras introduced in Jose’s talk. We start by recalling Procesi bundles, quantum Hamiltonian reductions, and Cherednik algebras. Then we apply Procesi bundles to relating the spherical Rational Cherednik algebras to quantum Hamiltonian reductions. Finally, we study the def...