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Double affine Hecke algebras and Calogero-Moser spaces


Author: Alexei Oblomkov
Journal: Represent. Theory 8 (2004), 243-266
MSC (2000): Primary 13C14, 15A27, 16H05
DOI: https://doi.org/10.1090/S1088-4165-04-00246-8
Published electronically: June 2, 2004
MathSciNet review: 2077482
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Abstract: In this paper we prove that the spherical subalgebra $eH_{1,\tau}e$ of the double affine Hecke algebra $H_{1,\tau}$ is an integral Cohen-Macaulay algebra isomorphic to the center $Z$ of $H_{1,\tau}$, and $H_{1,\tau}e$ is a Cohen-Macaulay $eH_{1,\tau}e$-module with the property $H_{1,\tau}=\operatorname{End}_{eH_{1,\tau}e}(H_{1,\tau}e)$ when $\tau$ is not a root of unity. In the case of the root system $A_{n-1}$ the variety $\operatorname{Spec}(Z)$ is smooth and coincides with the completion of the configuration space of the Ruijenaars-Schneider system. It implies that the module $eH_{1,\tau}$ is projective and all irreducible finite dimensional representations of $H_{1,\tau}$ are isomorphic to the regular representation of the finite Hecke algebra.


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Additional Information

Alexei Oblomkov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S1088-4165-04-00246-8
Received by editor(s): July 10, 2003
Received by editor(s) in revised form: April 26, 2004
Published electronically: June 2, 2004
Additional Notes: This work was partially supported by the NSF grant DMS-9988796
Article copyright: © Copyright 2004 American Mathematical Society

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