### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Overview

Double affine Hecke algebras, also called Cherednik algebras, were introduced by Cherednik in 1993 as a tool in his proof of Macdonald’s conjectures about orthogonal polynomials for root systems. Since then, it has been realized that Cherednik algebras are of great independent interest; they appeared in many different mathematical contexts and found many applications.

The goal of this course is to give an introduction to Cherednik algebras, and to review the web of connections between them and other mathematical objects. For this reason, the course consists of several parts that are relatively independent of each other. Also, because of such a wide scope, many important topics are skipped and many proofs are omitted or sketched; we explain in detail only those proofs which can be understood in class and are instructive. We also try to focus on explicit examples.

### Grading

Course grade is based upon class attendance and participation. There are no homework assignments, projects, or exams.

### Calendar

WEEK # | TOPICS |
---|---|

1 | Classical and quantum Olshanetsky-Perelomov systems for finite Coxeter groups |

2 | The rational Cherednik algebra I |

3 |
The rational Cherednik algebra II Finite Coxeter groups and the Macdonald-Mehta integral |

4 | The Macdonald-Mehta integral |

5 | Parabolic induction and restriction functors for rational Cherednik algebras |

6 |
The Knizknik-Zamolodchikov functor Rational Cherednik algebras for varieties with group actions |

7 | Hecke algebras for varieties with group actions |

8 | Symplectic reflection algebras I |

9 | Symplectic reflection algebras II |

10 | Calogero-Moser spaces |

11 | Quantization of Calogero-Moser spaces |