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
Lecture 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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