Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products
Publications Mathématiques de l'IHÉS, Tome 105 (2007), pp. 91-155.

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

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     title = {Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {91--155},
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     year = {2007},
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     zbl = {1188.16010},
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     url = {http://www.numdam.org/articles/10.1007/s10240-007-0005-9/}
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Etingof, Pavel; Gan, Wee Liang; Ginzburg, Victor; Oblomkov, Alexei. Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products. Publications Mathématiques de l'IHÉS, Tome 105 (2007), pp. 91-155. doi : 10.1007/s10240-007-0005-9. http://www.numdam.org/articles/10.1007/s10240-007-0005-9/

1. A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, part 1, pp. 1-50, Amer. Math. Soc., Providence, RI, 1993. | MR | Zbl

2. Y. Berest, P. Etingof, V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J., 118 (2003), 279-337 | MR | Zbl

3. R. Bezrukavnikov, M. Finkelberg, V. Ginzburg, with an Appendix by P. Etingof, Cherednik algebras and Hilbert schemes in characteristic p , Represent. Theory, 10 (2006), 254-298 | MR | Zbl

4. M. Boyarchenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math., 211 (2007), 244-265 | MR

5. W. Crawley-Boevey, Decomposition of Marsden-Weinstein reductions for representations of quivers, Compos. Math., 130 (2002), 225-239 | MR | Zbl

6. W. Crawley-Boevey, M.P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605-635 | MR | Zbl

7. C. Dunkl, E. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc., 86 (2003), 70-108 | MR | Zbl

8. P. Etingof, Cherednik and Hecke algebras of varieties with a finite group action, preprint. math.QA/0406499.

9. P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math., 147 (2002), 243-348 | MR | Zbl

10. P. Etingof, V. Ginzburg and E. Rains, in preparation.

11. W.L. Gan, Reflection functors and symplectic reflection algebras for wreath products, Adv. Math., 205 (2006), 599-630 | MR

12. W.L. Gan, V. Ginzburg, Deformed preprojective algebras and symplectic reflection algebras for wreath products, J. Algebra, 283 (2005), 350-363 | MR

13. W. L. Gan and V. Ginzburg, Almost-commuting variety, D-modules, and Cherednik algebras, IMPR, Int. Math. Res. Pap., 2006 (2006), Article ID 26439. math.RT/0409262. | MR | Zbl

14. I. Gordon, A remark on rational Cherednik algebras and differential operators on the cyclic quiver, Glasg. Math. J., 48 (2006), 145-160 | MR

15. I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes I, II, Adv. Math., 198 (2005), 222-274 and Duke Math. J., 132 (2006), 73-135. math.RA/0407516 and math.RT/0410293. | MR | Zbl

16. N. Guay, Cherednik algebras and Yangians, Int. Math. Res. Not., 2005 (2005), 3551-3593 | MR | Zbl

17. N. Guay, Affine Yangians and deformed double current algebras in type A, Adv. Math., 211 (2007), 436-484 | MR

18. N. Guay, Quantum algebras and symplectic reflection algebras for wreath products, preprint.

19. M.P. Holland, Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers, Ann. Sci. Éc. Norm. Supér., IV. Sér., 32 (1999), 813-834 | Numdam | MR | Zbl

20. P.B. Kronheimer, The construction of ALE spaces as hyper-Kahler quotients, J. Differ. Geom., 29 (1989), 665-683 | MR | Zbl

21. G. Lusztig, Quivers, Perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4 (1991), 365-421 | MR | Zbl

22. G. Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier, 50 (2000), 461-489 | Numdam | MR | Zbl

23. A. Maffei, A remark on quiver varieties and Weyl groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 1 (2002), 649-686 | Numdam | MR

24. A. Mellit, Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation, Algebra Discrete Math., 2004 (2004), 89-110 | MR | Zbl

25. S. Montarani, Finite dimensional representations of symplectic reflection algebras associated to wreath products II, preprint. math.RT/0501156. | MR

26. I. Musson, Hilbert schemes and noncommutative deformations of type A Kleinian singularities, J. Algebra, 293 (2005), 102-129 | MR | Zbl

27. H. Nakajima, Reflection functors for quiver varieties and Weyl group actions, Math. Ann., 327 (2003), 671-721 | MR | Zbl

28. A. Oblomkov, Deformed Harish-Chandra homomorphism for the cyclic quiver, preprint. math.RT/0504395. | MR

29. C. Ringel, The rational invariants of the tame quivers, Invent. Math., 58 (1980), 217-239 | MR | Zbl

30. J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., 36 (1976), 295-312 | MR | Zbl

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