Cherednik algebras and differential operators on quasi-invariants

TL;DR

This paper explores the representation theory of rational Cherednik algebras related to Coxeter groups, demonstrating their simplicity and Morita equivalence to differential operator algebras, and investigates the structure of differential operators on quasi-invariants.

Contribution

It establishes the simplicity and Morita equivalence of Cherednik algebras for integral parameters and analyzes the algebra of differential operators on quasi-invariants, revealing their structure and relations.

Findings

H is simple and Morita equivalent to D(h)#W for integral c

D(Q) is simple and Morita equivalent to D(h)

D(Q)^W is isomorphic to the spherical subalgebra eHe

Abstract

We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. We further study an algebra Q of quasi-invariant polynomials on h introduced by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset C[h]. We prove that the algebra D(Q) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra, by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective (Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).