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 |
Disability Support Services: If you have a physical, psychological, medical, or learning disability that may affect your course work, please contact Disability Support Services (DSS) office: ECC (Educational Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with you what accommodations are necessary and appropriate. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated. All information and documentation of disability is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and DSS. For procedures and information, go to the following web site http://www.ehs.sunysb.edu and search Fire safety and Evacuation and Disabilities.
Academic Integrity: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instance of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website athttp://www.stonybrook.edu/uaa/academicjudiciary/.
Critical Incident Management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, and/or inhibits students' ability to learn.