Double affine Hecke algebras and Calogero-Moser spaces

Abstract

In this paper we prove that the spherical subalgebra $eH_{1,τ}e$ of the double affine Hecke algebra $H_{1,τ}$ is an integral Cohen-Macaulay algebra isomorphic to the center $Z$ of $H_{1,τ}$, and $H_{1,τ}e$ is a Cohen-Macaulay $eH_{1,τ}e$-module with the property $H_{1,τ}=End_{eH_{1,τ}e}(H_{1,τ}e)$. In the case of the root system $A_{n-1}$ the variety $Spec(Z)$ is smooth and coincides with the completion of the configuration space of the relativistic analog of the trigomonetric Calogero-Moser system. This implies the result of Cherednik that the module $eH_{1,τ}$ is projective and all irreducible finite dimensional representations of $H_{1,τ}$ are regular representation of the finite Hecke algebra.