We give a new version of the fusion procedure for the symmetric group which originated in the work of Jucys and was developed by Cherednik. We derive it from the Jucys–Murphy formulas for the diagonal matrix units for the symmetric group.

We show how the knowledge of the Fourier coefficients of the Cherednik kernel leads to combinatorial formulas for generalized exponents. We recover known formulas for generalized exponents of irreducible representations parameterized by dominant roots, and obtain new formulas for the generalized exponents for irreducible representations parameterized by the dominant elements of the root lattice...

Recently Delorme and Opdam have generalized the theory of Rgroups towards affine Hecke algebras with unequal labels. We apply their results in the case where the affine Hecke algebra is of type B, for an induced discrete series representation with real central character. We calculate the R-group of such an induced representation, and show that it decomposes multiplicity free into 2 irreducible ...

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 goal of this paper is to define a new class of objects which we call triple groups and to relate them with Cherednik's double affine Heeke algebras. This has as immediate consequences new descriptions of double affine Weyl and Artin groups, the double affine Heeke algebras as well as the corresponding elliptic objects. From the new descriptions we recover results of Cherednik on automorphis...

The goal of this talk is to prove an analog of the Beilinson-Bernstein localization theorem for Cherednik algebras. Strictly speaking we will only do this for categories O, in fact, the localization theorem for all modules follows from here. Let us recall the notation and some definitions. We consider the reflection representation h of the symmetric group Sn. By X we denote the “normalized” Hil...

We construct a microlocalization of the rational Cherednik algebras H of type Sn. This is achieved by a quantization of the Hilbert scheme Hilb C2 of n points in C2. We then prove the equivalence of the category of H -modules and that of modules over its microlocalization under certain conditions on the parameter.