Double affine Hecke algebras and Calogero-Moser spaces

Oblomkov, Alexei

doi:10.1090/S1088-4165-04-00246-8

Double affine Hecke algebras and Calogero-Moser spaces
HTML articles powered by AMS MathViewer

by Alexei Oblomkov

Represent. Theory 8 (2004), 243-266

DOI: https://doi.org/10.1090/S1088-4165-04-00246-8

Published electronically: June 2, 2004

PDF | Request permission

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.

References

Ivan Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators, Internat. Math. Res. Notices 9 (1992), 171–180. MR 1185831, DOI 10.1155/S1073792892000199
Ivan Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), no. 1, 119–145. MR 1354956, DOI 10.1007/BF01231441
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
Kyoji Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, 75–179. MR 780892, DOI 10.2977/prims/1195179841
Kyoji Saito and Tadayoshi Takebayashi, Extended affine root systems. III. Elliptic Weyl groups, Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 301–329. MR 1442503, DOI 10.2977/prims/1195145453
Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), no. 4, 481–507. MR 478225, DOI 10.1002/cpa.3160310405
Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344, DOI 10.1090/ulect/018
George Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), no. 1, 1–41. With an appendix by I. G. Macdonald. MR 1626461, DOI 10.1007/s002220050237
Yuri Berest and George Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry, Int. Math. Res. Not. 26 (2002), 1347–1396. With an appendix by Michel Van den Bergh. MR 1904791, DOI 10.1155/S1073792802108051
S. N. M. Ruijsenaars and H. Schneider, A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), no. 2, 370–405. MR 851627, DOI 10.1016/0003-4916(86)90097-7
V. V. Fock and A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, vol. 191, Amer. Math. Soc., Providence, RI, 1999, pp. 67–86. MR 1730456, DOI 10.1090/trans2/191/03
A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken, Quasi-Poisson manifolds, Canad. J. Math. 54 (2002), no. 1, 3–29. MR 1880957, DOI 10.4153/CJM-2002-001-5
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
Alexander A. Kirillov Jr., Lectures on affine Hecke algebras and Macdonald’s conjectures, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 251–292. MR 1441642, DOI 10.1090/S0273-0979-97-00727-1
Siddhartha Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), no. 1, 267–282. MR 1715325, DOI 10.2307/121102
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468, DOI 10.1007/978-3-662-21576-0
Bogdan Ion, Involutions of double affine Hecke algebras, Compositio Math. 139 (2003), no. 1, 67–84. MR 2024965, DOI 10.1023/B:COMP.0000005078.39268.8d
Robert Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177. MR 372897, DOI 10.1016/0040-9383(75)90025-7
Joseph Bernstein, Alexander Braverman, and Dennis Gaitsgory, The Cohen-Macaulay property of the category of $(\mathfrak {g},K)$-modules, Selecta Math. (N.S.) 3 (1997), no. 3, 303–314. MR 1481131, DOI 10.1007/s000290050012
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833, DOI 10.1007/978-3-642-57908-0
Ivan Cherednik, Diagonal coinvariants and double affine Hecke algebras, Int. Math. Res. Not. 16 (2004), 769–791. MR 2036955, DOI 10.1155/S1073792804131577
M. A. Olshanetsky and A. M. Perelomov, Integrable systems and Lie algebras, Mathematical physics reviews, Vol. 3, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 3, Harwood Academic, Chur, 1982, pp. 151–220. MR 704030, DOI 10.1007/BF01231441