Cherednik algebras and differential operators on quasi-invariants

Abstract

We develop representation theory of the rational Cherednik algebra ${\rm H}\sb c$ associated to a finite Coxeter group $W$ in a vector space $\mathfrak {h}$, and a parameter "c." We use it to show that, for integral values of "c," the algebra ${\rm H}\sb c$ is simple and Morita equivalent to $\mathscr {D}(\mathfrak {h})\#W$, the cross product of $W$ with the algebra of polynomial differential operators on $\mathfrak {h}$.

O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, $Q\sb c$, of quasi-invariant polynomials on $\mathfrak {h}$, such that $\mathbb {C}[\mathfrak {h}]\sp W\subset Q\sb c\subset \mathbb {C}[\mathfrak {h}]$. We prove that the algebra $\mathscr {D}(Q\sb c)$ of differential operators on quasi-invariants is a simple algebra, Morita equivalent to $\mathscr {D}(\mathfrak {h})$. The subalgebra $\mathscr {D}(Q\sb c)\sp W\subset \mathscr {D}(Q\sb c)$ of $W$-invariant operators turns out to be isomorphic to the spherical subalgebra $\mathbf {eH}\sb c\mathbf {e}\subset {\rm H}\sb c$. We show that $\mathscr {D}(Q\sb c)$ is generated, as an algebra, by $Q\sb c$ and its "Fourier dual" $Q\sb c\sp \flat$, and that $\mathscr {D}(Q\sb c)$ is a rank-one projective $(Q\sb c\otimes Q\sb c\sp \flat)$-module (via multiplication-action on $\mathscr {D}(Q\sb c)$ on opposite sides).