Cherednik Algebras and DAHAs
The course is devoted to two families of algebras actively studied over the past two decades: the double affine Hecke algebras (DAHAs) introduced by Ivan Cherednik, and their rational version, the so-called Cherednik algebras introduced by Pavel Etingof and Victor Ginzburg. We will describe the foundations of their theory and discuss some applications and links - mostly, to integrable systems but also touching upon orthogonal polynomials, combinatorics, and geometry. To some extent, similar constructions exist at the elliptic level as well; these will be discussed towards the end of the course.
Lecturer
Oleg Chalykh
Date
3rd March ~ 28th April, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 13:30 - 15:05 | A3-4-101 | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
Basic notions of linear algebra and representation theory. Some knowledge of Coxeter groups, simple complex Lie algebras and root systems is desirable but not essential as all the necessary concepts will be introduced in the course. Some knowledge of integrable systems would be useful but is not required.
Syllabus
1. Background material: Coxeter groups and Hecke algebras, root systems and Dynkin diagrams, classical and quantum integrable systems.
2. Rational Cherednik algebras and Dunkl operators. Category O and KZ equations. Applications: Calogero-Moser systems; diagonal harmonics.
3. Affine Weyl groups and affine Hecke algebras. Double affine Hecke algebras and Cherednik operators.
4. Macdonald-Ruijsenaars systems, Macdonald polynomials, Macdonald symmetric functions.
5. Star-shaped affine Dynkin quivers, generalised DAHAs, and character varieties.
6. Global Cherednik algebras. Elliptic Cherednik algebras and elliptic Dunkl operators. Applications: elliptic Calogero-Moser system, Inozemtsev spin chain.
7. Elliptic Cherednik operators and Ruijsenaars systems.
2. Rational Cherednik algebras and Dunkl operators. Category O and KZ equations. Applications: Calogero-Moser systems; diagonal harmonics.
3. Affine Weyl groups and affine Hecke algebras. Double affine Hecke algebras and Cherednik operators.
4. Macdonald-Ruijsenaars systems, Macdonald polynomials, Macdonald symmetric functions.
5. Star-shaped affine Dynkin quivers, generalised DAHAs, and character varieties.
6. Global Cherednik algebras. Elliptic Cherednik algebras and elliptic Dunkl operators. Applications: elliptic Calogero-Moser system, Inozemtsev spin chain.
7. Elliptic Cherednik operators and Ruijsenaars systems.
Reference
1. Etingof, P.: Calogero-Moser systems and representation theory. Vol. 4. European Mathematical Society, 2007.
2. Chalykh, O.: Dunkl and Cherednik Operators. Encyclopedia of Mathematical Physics (Second edition), vol. 3, 309-327. Academic Press, 2025.
3. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243-348, 2002.
4. Kirillov Jr, A.: Lectures on affine Hecke algebras and Macdonald’s conjectures. Bull. AMS 34, no. 3, 251-292, 1997.
5. Macdonald, I. G.: Affine Hecke algebras and orthogonal polynomials. No. 157. Cambridge University Press, 2003.
6. Cherednik, I.: Double affine Hecke algebras. Vol. 319. Cambridge University Press, 2005.
Background references on Coxeter groups and root systems:
7. Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press, 1990.
8. Bourbaki, N.: Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6 (English translation). Springer, 2002.
2. Chalykh, O.: Dunkl and Cherednik Operators. Encyclopedia of Mathematical Physics (Second edition), vol. 3, 309-327. Academic Press, 2025.
3. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math. 147, 243-348, 2002.
4. Kirillov Jr, A.: Lectures on affine Hecke algebras and Macdonald’s conjectures. Bull. AMS 34, no. 3, 251-292, 1997.
5. Macdonald, I. G.: Affine Hecke algebras and orthogonal polynomials. No. 157. Cambridge University Press, 2003.
6. Cherednik, I.: Double affine Hecke algebras. Vol. 319. Cambridge University Press, 2005.
Background references on Coxeter groups and root systems:
7. Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press, 1990.
8. Bourbaki, N.: Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6 (English translation). Springer, 2002.
Video Public
Yes
Notes Public
Yes
Language
English