We conjecture that the “nilpotent points” of Calogero-Moser space for reflection groups are parametrised naturally by the two-sided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the case G = μ` o Sn and show that in type B our description produces an existing conjectural description of two-s...

We present new examples of deformations of smash product algebras that arise from Hopf algebra actions on pairs of module algebras. These examples involve module algebras that are Koszul, in which case a PBW theorem we established previously applies. Our construction generalizes several ‘double’ constructions appearing in the literature, including Weyl algebras and some types of Cherednik algeb...

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBWtype basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We ...

Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers–Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and explain all of them by showing the connection with Y -systems associated with (untwisted and twiste...

In type A, the q, t-Fuß-Catalan numbers Cat n (q, t) can be defined as a bigraded Hilbert series of a module associated to the symmetric group Sn. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to some...

Using combinatorial properties of complex reflection groups we show that if the group W is different from the wreath product Sn ≀ Z/mZ and the binary tetrahedral group (labelled G(m, 1, n) and G4 respectively in the Shephard-Todd classification), then the generalised Calogero-Moser space Xc associated to the centre of the rational Cherednik algebra H0,c(W ) is singular for all values of the par...

In type A, the q, t-Fuß-Catalan numbers Cat (m) n (q, t) can be defined as a bigraded Hilbert series of a module associated to the symmetric group Sn. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to ...

We provide Harish-Chandra type formulas for the multivariate Bessel functions and HeckmanOpdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. in [EK94]. Integration over the Liouville tori of the Gelf...

in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

In this paper we prove the existence of the Dunkl weight function Kc,λ for any irreducible representation λ of any finite Coxeter group W , generalizing previous results of Dunkl. In particular, Kc,λ is a family of tempered distributions on the real reflection representation of W taking values in EndC(λ), with holomorphic dependence on the complex multi-parameter c. When the parameter c is real...