Let R be a root system of type BC in a = Rr of general positive multiplicity. We introduce certain canonical weight function on Rr which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and ...

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...

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...

We introduce an algebra H consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type (C ∨ 1 , C1). We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra H, and the corresponding Fourier transforms. These eigenfunctions are non-symmetric v...

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...

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic We...

The finite XXZ spin chain with boundaries is studied. We derive the transfer matrix from the q-difference equation discovered by Cherednik and construct its eigenstates by the vertex operator approach. We point out that the eigenstates with no magnetic fields have a symmetry called the turning symmetry. Making use of this symmetry we calculate the spontaneous magnetization in the thermodynamic ...

For any algebraic curve C and n ≥ 1, P. Etingof introduced a ‘global’ Cherednik algebra as a natural deformation of the cross product D(Cn)⋊Sn, of the algebra of differential operators on Cn and the symmetric group. We provide a construction of the global Cherednik algebra in terms of quantum Hamiltonian reduction. We study a category of character Dmodules on a representation scheme associated ...