We study multilevel matrix ensembles at general β by identifying them with a class of processes defined via the branching rules for multivariate Bessel and Heckman-Opdam hypergeometric functions. For β = 1, 2, we express the joint multilevel density of the eigenvalues of a generalized Wishart matrix as a multivariate Bessel ensemble, generalizing a result of Dieker-Warren in [DW09]. In the null...

The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous non-symmetric eigenfunctions of the AN−1 Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomial...

The Jack polynomials Jλ(x; α) form a remarkable class of symmetric polynomials. They are indexed by a partition λ and depend on a parameter α. One of their properties is that several classical families of symmetric functions can be obtained by specializing α, e.g., the monomial symmetric functions mλ (α = ∞), the elementary functions eλ′ (α = 0), the Schur functions sλ (α = 1) and finally the t...

We study the endomorphism algebras of Verma modules for rational Cherednik algebras at t = 0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvarieties of the generalized Calogero–Moser space. In the introduction, we motivate our results by describing them in the context of deriv...

Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of differential operators on h). These algebras are generated by G, h, h with certain commutation relations, ...

The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous non-symmetric eigenfunctions of the AN−1 Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomial...

Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between rational Calogero-Moser system with harmonic term its trigonometric version. We present conceptual explanation of this using Cherednik algebra establish quasi-invariant extension. More specifically, we consider configurations $\mathcal A$ real hyperplanes multiplicities admitting Baker-Akhiezer function use to int...