**MAT
536. Algebra III - CHEREDNIK ALGEBRAS**

Mikhail
Movshev,
Fall 2014, Stony Brook University

TuTh 2:30pm - 3:50pm, in
Earth&Space 181

__First
class: Tu 9/4__

**Overview:** The
goal of the corse is to give an introduction to Cherednik algebras,
and

to review the web of connections between them and other mathematical objects.

**Grading:** Please
see the instructor if you have not passed your orals yet.

**Textbook:**

P.
Etingof and X. Ma *L**ecture
Notes On Cherednik Algebras*

**Schedule:**

Date |
Topic |

09/02/14 |
No class |

09/04/14 |
Introduction |

09/09/14 |
The rational quantum Calogero-Moser system Complex reflection groups Olshanetsky-Perelomov operators |

09/11/14 |
Dunkl operators Proof of Theorem 2.9 Uniqueness of the operators Lj Classical Dunkl operators and Olshanetsky-Perelomov Hamiltonians |

09/16/14 |
Rees algebras The rational Cherednik algebra,Definition and examples The PBW theorem for the rational Cherednik algebra The spherical subalgebra |

09/18/14 |
The localization lemma Category O for rational Cherednik algebras The grading element Standard modules Finite length Characters |

09/23/14 |
Irreducible modules The contragredient module The contravariant form The matrix of multiplicities Example: the rank 1 case The Frobenius property Representations of H1,c of type A |

09/25/14 |
The Macdonald-Mehta integral Proof of Theorem 4.1 |

09/30/14 |
Application: the supports of Lc(C) |

10/02/14 |
Parabolic induction and restriction functors for rational Cherednik algebras A geometric approach to rational Cherednik algebras Completion of rational Cherednik algebras |

10/07/14 |
The duality functor Generalized Jacquet functors The centralizer construction Completion of rational Cherednik algebras at arbitrary points of h/G The completion functor |

10/09/14 |
Parabolic induction and restriction functors for rational Cherednik algebras Some evaluations of the parabolic induction and restriction functors Dependence of the functor Resb on b Supports of modules |

10/14/14 |
The Knizhnik-Zamolodchikov functor Braid groups and Hecke algebras KZ functors The image of the KZ functor |

10/16/14 |
Example: the symmetric group Sn Rational Cherednik algebras and Hecke algebras for varieties with group actions Twisted differential operators Some algebraic geometry preliminaries The Cherednik algebra of a variety with a finite group action Globalization |

10/21/14 |
Modified Cherednik algebra Orbifold Hecke algebras Hecke algebras attached to Fuchsian groups Hecke algebras of wallpaper groups and del Pezzo surfaces The Knizhnik-Zamolodchikov functor |

10/23/14 |
Proof of Theorem 7.15 Example: the simplest case of double affine Hecke algebras Affine and extended affine Weyl groups Cherednik’s double affine Hecke algebra of a root system Algebraic flatness of Hecke algebras of polygonal Fuchsian groups |

10/28/14 |
No class |

10/30/14 |
Symplectic reflection algebras The definition of symplectic reflection algebras The PBW theorem for symplectic reflection algebras Koszul algebras Proof of Theorem 8.6 |

11/04/14 |
The spherical subalgebra of the symplectic reflection algebra The center of the symplectic reflection algebra Ht,c . A review of deformation theory |

11/06/14 |
Deformation-theoretic interpretation of symplectic reflection algebras Finite dimensional representations of H0,c . |

11/11/14 |
Azumaya algebras Cohen-Macaulay property and homological dimension Proof of Theorem 8.32 |

11/13/14 |
Calogero-Moser spaces Hamiltonian reduction along an orbit The Calogero-Moser space The Calogero-Moser integrable system Proof of Wilson’s theorem |

11/18/14 |
The Gan-Ginzburg theorem The space Mc for Sn and the Calogero-Moser space. The Hilbert scheme Hilbn(C2) and the Calogero-Moser space The cohomology of Cn |

11/20/14 |
Quantization of Claogero-Moser spaces Quantum moment maps and quantum Hamiltonian reduction The Levasseur-Stafford theorem |

11/25/14 |
Corollaries of Theorem 10.1 |

11/27/14 |
The deformed Harish-Chandra homomorphism |

12/02/14 |
The deformed Harish-Chandra homomorphism |

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