University: Massachusetts Institute of Technology
Instructors: Prof. Pavel Etingof
Course Number: 18.735
Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.
The course notes were prepared jointly by Prof. Pavel Etingof and Xiaoguang Ma.
In order to prepare for class, students are required to read selections from the course notes. These readings can be found on the lecture notes page.
Course readings. WEEK # TOPICS READINGS 1 Classical and quantum Olshanetsky-Perelomov systems for finite Coxeter groups Chapter 2 2 The rational Cherednik algebra I Chapter 3, sections 3.1-3.13 3
The rational Cherednik algebra II
Finite Coxeter groups and the Macdonald-Mehta integral
Chapter 3, sections 3.14-3.17
Chapter 4, section 4.1
4 The Macdonald-Mehta integral Chapter 4, sections 4.2-4.4 5 Parabolic induction and restriction functors for rational Cherednik algebras Chapter 5 6
The Knizknik-Zamolodchikov functor
Rational Cherednik algebras for varieties with group actions
Chapter 7, sections 7.1-7.5
7 Hecke algebras for varieties with group actions Chapter 7, sections 7.6-7.15 8 Symplectic reflection algebras I Chapter 8, sections 8.1-8.7 9 Symplectic reflection algebras II Chapter 8, sections 8.8-8.13 10 Calogero-Moser spaces Chapter 9 11 Quantization of Calogero-Moser spaces Chapter 10
Bezrukavnikov, R., and P. Etingof. "Parabolic Induction and Restriction Functors for Rational Cherednik Algebras." Selecta Math 14, nos. 3-5 (2009): 397-425.
Etingof, P., and V. Ginzburg. "Symplectic Reflection Algebras, Calogero-Moser Space, and Deformed Harish-Chandra Homomorphism." arXiv:math/0011114.
Rouquier, R. "Representations of Rational Cherednik Algebras." arXiv:math/0504600.
Etingof, P. Lectures on Calogero-Moser Systems. arXiv:math/0606233.
———. "Cherednik and Hecke Algebras of Varieties With a Finite Group Action." arXiv: math.QA/0406499.
———. "A Uniform Proof of the Macdonald-Mehta-Opdam Identity for Finite Coxeter Groups." arXiv:0903.5084.
———. "Supports of Irreducible Spherical Representations of Rational Cherednik Algebras of Finite Coxeter Groups." arXiv:0911.3208.