We construct a microlocalization of the rational Cherednik algebras $H$ of type $S_n$. This is achieved by a quantization of the Hilbert scheme $\mathrm{Hilb}^n{\mathbb C}^2$ of $n$ points in ${\mathbb C}^2$. We then prove the equivalence of the category of $H$-modules and that of modules over its microlocalization under certain conditions on the parameter
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
References
A. BeĭLinson and J. Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15--18.
Y. Berest, P. Etingof, and V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279--337.
R. Bezrukavnikov and P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, preprint,\arxiv0803.3639v1[math.RT]
R. Bezrukavnikov, M. Finkelberg, and V. Ginzburg, Cherednik algebras and Hilbert schemes in characteristic $p$, with an appendix by P. Etingof, Represent. Theory 10 (2006), 254--298.
N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4--6, Elem. Math. (Berlin), Springer, Berlin, 2002.
P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243--348.
W. L. Gan and V. Ginzburg, Almost-commuting variety, $\mathscrD$-modules, and Cherednik algebras, with an appendix by V. Ginzburg, IMRP Int. Math. Res. Pap. 2006, no. 26439.
I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), 222--274.
—, Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves, Duke Math. J. 132 (2006), 73--135.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, 2nd ed., Cambridge Univ. Press, Cambridge, 1990.
M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941--1006.
—, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371--407.
G. J. Heckman, ``A remark on the Dunkl differential-difference operators'' in Harmonic Analysis on Reductive Groups (Brunswick, Me., 1989), Progr. Math. 101, Birkhäuser, Boston, 1991, 181--191.
R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), 327--358.
D. Kaledin, Geometry and topology of symplectic resolutions, preprint,\arxivmath/0608143v1[math.AG]
M. Kashiwara, ``Representation theory and $D$-modules on flag varieties'' in Orbites unipotentes et représentations, III, Astérisque 173 --.174, Soc. Math. France, Montrouge, 1989, 55--109.
—, Quantization of contact manifolds, Publ. Res. Inst. Math. Sci. 32 (1996), 1--7.
—, $D$-Modules and Microlocal Calculus, Transl. Math. Monogr. 217, Amer. Math. Soc., Providence, 2003.
—, ``Equivariant derived category and representation of real semisimple Lie groups'' in Representation Theory and Complex Analysis,(Venice, 2004), Lecture Notes in Math. 1931, Springer, Berlin, 2008, 137--234.
M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations, III: Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), 813--979.
D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481--507.
M. Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), 271--294.
H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
P. Polesello and P. Schapira, Stacks of quantization-deformation modules on complex symplectic manifolds, Int. Math. Res. Not. 2004, no. 49, 2637--2664.