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On the geometry of the Calogero-Moser system. (English) Zbl 1130.33006

The authors carry over their previous work on geometric structures on complements of projectivized hyperplane arrangements to the projective structures on certain spaces of conjugacy classes in semisimple complex Lie groups. These projective structures are obtained from a study of the quantized periodic Calogero-Moser system associated with a root system. Under some constraints these projective structures underlie a complex hyperbolic structure, which is a ball quotient minus a Heegner divisor if a certain Schwarz-type condition is satisfied. The authors discuss an example related to the moduli spaces of degree 12 divisors on the projective line. They also propose thorough a study of Calogero-Moser systems associated with arbitrary root systems and spectral/coupling parameters.

MSC:

33C67 Hypergeometric functions associated with root systems
14J15 Moduli, classification: analytic theory; relations with modular forms
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[1] Allcock, D.; Carlson, J.; Toledo, D., The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Alg. Geom., 11, 659-724 (2002) · Zbl 1080.14532
[2] Beilinson, A.; Drinfeld, V., Quantization of Hitchin’s fibration and Langlands program, Math. Phys. Stud., 19, 3-7 (1996) · Zbl 0864.14007
[3] Beilinson, A.; Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigensheaves (1997), Preprint
[4] Ben-Zvi D., Nevins T. — From solitons to many body systems, math.AG/0310490.; Ben-Zvi D., Nevins T. — From solitons to many body systems, math.AG/0310490. · Zbl 1153.37031
[5] Bourbaki, N., Groupes et algèbres de Lie (1968), Chapitres 4, 5 et 6 Masson: Chapitres 4, 5 et 6 Masson Paris · Zbl 0186.33001
[6] Cherednik, I., A unification of Knizhnik-Zamolodchikov equations and Dunkl operators via affine Hecke algebras, Invent. Math., 106, 411-432 (1991) · Zbl 0725.20012
[7] Couwenberg W, Heckman G., Looijenga E. — Geometric structures on the complement of a projective arrangement, math.AG/0311404.; Couwenberg W, Heckman G., Looijenga E. — Geometric structures on the complement of a projective arrangement, math.AG/0311404. · Zbl 1083.14039
[8] Deligne, P.; Mostow, D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHES, 63, 58-89 (1986) · Zbl 0615.22008
[9] Drinfeld, V. G., Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl., 20, 58-60 (1986) · Zbl 0599.20049
[10] Heckman, G. J., An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math., 103, 341-350 (1991) · Zbl 0721.33009
[11] Heckman, G. J., Dunkl operators, (Séminaire Bourbaki 828. Séminaire Bourbaki 828, Astérisque, 245 (1997)), 223-246 · Zbl 0916.33012
[12] Heckman, G.; Looijenga, E., The moduli space of rational elliptic surfaces, Alg. Geom. 2000 Azumino, Adv. Stud. Pure Math., 36, 185-248 (2002) · Zbl 1063.14044
[13] Heckman, G. J.; Opdam, E. M., Root systems and hypergeometric functions IV (by O), Comp. Math., 67, 191-209 (1988) · Zbl 0669.33008
[14] [14] Hurtibise J.C., Markman E. — Calogero-Mosey systems and Hitchin systems, math. AG/9912161.; [14] Hurtibise J.C., Markman E. — Calogero-Mosey systems and Hitchin systems, math. AG/9912161.
[15] Kondo, S., A complex hyperbolic structure on the moduli space of curves of genus three, J. Reine u. Angew. Math., 525, 219-232 (2000) · Zbl 0990.14007
[16] Looijenga E. — Affine Artin groups and the fundamental groups of some moduli spaces, math. AG/9801117.; Looijenga E. — Affine Artin groups and the fundamental groups of some moduli spaces, math. AG/9801117.
[17] Macdonald, I. G., Affine Hecke Algebras and Orthogonal Polynomials, (Cambridge Tracts in Math., vol. 157 (2003), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0883.33008
[18] Matsuo, A., Integrable connections related to zonal spherical functions, Invent. Math., 110, 95-121 (1992) · Zbl 0801.35131
[19] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformation, Adv. in Math., 16, 197-220 (1975) · Zbl 0303.34019
[20] Olshanetsky, M. A.; Perelomov, A. M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math., 37, 93-108 (1976) · Zbl 0342.58017
[21] Olshanetsky, M. A.; Perelomov, A. M., Classical integrable finite dimensional systems related to Lie algebras, Phys. Rep., 71, 5, 313-400 (1981)
[22] Olshanetsky, M. A.; Perelomov, A. M., Quantum integrable systems related to Lie algebras, Phys. Rep., 91, 6, 314-403 (1983)
[23] Opdam, E. M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175, 75-121 (1995) · Zbl 0836.43017
[24] Opdam, E. M., Lecture notes on Dunkl operators for real and complex reflection groups, Math. Soc. Japan Memoirs, 8 (2000) · Zbl 0984.33001
[25] Vakil, R., Twelve points on the projective line, branched covers, and rational elliptic fibrations, Math. Ann., 320, 1, 33-54 (2001) · Zbl 1017.14005
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