Graded Cherednik Algebra And Quasi-Invariant Differential Forms

Abstract

Given a finite Coxeter group W acting on its finite dimensional reflection representation V, we consider the ring D(ΩV)[RTIMES]W of equivariant differential operators on differential forms on V. After computing its Hochschild cohomology, we deduce the existence of a universal formal deformation of it, parametrized by the set of conjugation invariant functions on the set of reflections of W. We then introduce the graded Cherednik algebra, which represents an algebraic realization of that deformation. Using the deformed de Rham differential introduced by Dunkl and Opdam, we deform the standard Lie derivative operator and construct a Dunkl type embedding for the graded Cherednik algebra which extends the classical one. This embedding allows us to endow the graded Cherednik algebra with a natural differential. We study its cohomology and relate it to the singular polynomials of W. Finally, following the idea in the classical case, we investigate different notions of quasi-invariant differential forms and their relation to the spherical subalgebra of the graded Cherednik algebra.