Orthogonal Polynomials, Double Affine Hecke Algebras, and Quantum Character Varieties
This is an introductory course on the Macdonald theory of orthogonal polynomials, Double Affine Hecke Algebras (DAHA), and their role in modern mathematics and physics. The main goal of the course is to provide foundational knowledge of orthogonal polynomials, preparing students to engage with modern research literature in representation theory, algebraic combinatorics, and related fields.
We will begin with the classical theory of orthogonal polynomials in one variable and their q-analogues, situating them within the Askey scheme. This foundation leads naturally to the ring of symmetric functions, Schur polynomials, and their generalizations to the Hall-Littlewood and Macdonald polynomials. A central theme will be the connection between Macdonald polynomials and Dunkl operators, which arise in quantum many-body systems.
Building on this, we will explore Cherednik's nonsymmetric generalizations and introduce the Double Affine Hecke Algebra (DAHA). In the final part of the course, we will examine the deep connection between DAHA and the geometry of quantum character varieties, which elucidates the action of the $SL(2,\mathbb Z)$ by automorphisms of DAHA. This interplay between algebra, geometry, and physics will provide a glimpse into the deep structures underlying modern mathematical research.
We will begin with the classical theory of orthogonal polynomials in one variable and their q-analogues, situating them within the Askey scheme. This foundation leads naturally to the ring of symmetric functions, Schur polynomials, and their generalizations to the Hall-Littlewood and Macdonald polynomials. A central theme will be the connection between Macdonald polynomials and Dunkl operators, which arise in quantum many-body systems.
Building on this, we will explore Cherednik's nonsymmetric generalizations and introduce the Double Affine Hecke Algebra (DAHA). In the final part of the course, we will examine the deep connection between DAHA and the geometry of quantum character varieties, which elucidates the action of the $SL(2,\mathbb Z)$ by automorphisms of DAHA. This interplay between algebra, geometry, and physics will provide a glimpse into the deep structures underlying modern mathematical research.
Lecturer
Date
26th February ~ 11th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 09:50 - 12:15 | Shuangqing-C546 | ZOOM 09 | 230 432 7880 | BIMSA |
Prerequisite
Undergraduate Abstract Algebra (required); Lie Algebras (recommended).
Syllabus
1. Orthogonal polynomials in one variable and the Askey scheme.
1.1 Classical orthogonal polynomials and their history.
1.2 q-Orthogonal polynomials and difference equations.
1.3 Hypergeometric functions.
1.4 q-Combinatorics.
1.5 Askey scheme and classification.
2. Multivariate orthogonal polynomials and Symmetric functions.
2.1 Direct and inverse limits of rings.
2.2 The ring of symmetric functions.
2.3 Elementary symmetric functions, Complete Symmetric functions, and Power sums.
2.4 Schur functions.
2.5 Scalar product and orthogonal polynomials.
2.6 Hall-Littlewood and Macdonald symmetric functions.
3. Affine Root Systems and Hecke Algebras
3.1 Review of finite root systems and simple Lie Algebras.
3.2 Macdonald scalar product.
3.3 (Extended) Affine Weyl group.
3.4 Affine Braid groups and Hecke Algebras.
3.5 Macdonald operators
4. Double Affine Hecke Algebra
4.1 Elliptic braid group.
4.2 Generators and relations, $SL(2,\mathbb Z)$ action by automorphisms.
4.3 Representation in difference operators.
4.4 Spherical subalgebra
4.5 $C^\vee C_1$-DAHA
5. Applications to geometry
5.1 Character varieties of surface groups
5.2 Mapping Class Group
5.3 Quantum character varieties and DAHA
1.1 Classical orthogonal polynomials and their history.
1.2 q-Orthogonal polynomials and difference equations.
1.3 Hypergeometric functions.
1.4 q-Combinatorics.
1.5 Askey scheme and classification.
2. Multivariate orthogonal polynomials and Symmetric functions.
2.1 Direct and inverse limits of rings.
2.2 The ring of symmetric functions.
2.3 Elementary symmetric functions, Complete Symmetric functions, and Power sums.
2.4 Schur functions.
2.5 Scalar product and orthogonal polynomials.
2.6 Hall-Littlewood and Macdonald symmetric functions.
3. Affine Root Systems and Hecke Algebras
3.1 Review of finite root systems and simple Lie Algebras.
3.2 Macdonald scalar product.
3.3 (Extended) Affine Weyl group.
3.4 Affine Braid groups and Hecke Algebras.
3.5 Macdonald operators
4. Double Affine Hecke Algebra
4.1 Elliptic braid group.
4.2 Generators and relations, $SL(2,\mathbb Z)$ action by automorphisms.
4.3 Representation in difference operators.
4.4 Spherical subalgebra
4.5 $C^\vee C_1$-DAHA
5. Applications to geometry
5.1 Character varieties of surface groups
5.2 Mapping Class Group
5.3 Quantum character varieties and DAHA
Reference
1) I.G.Macdonald Symmetric Functions and Orthogonal Polynomials. Vol. 12. American Mathematical Soc., 1998
2) I.Cherednik Double Affine Hecke Algebras Vol. 319. Cambridge University Press, 2005.
2) I.Cherednik Double Affine Hecke Algebras Vol. 319. Cambridge University Press, 2005.
Video Public
Yes
Notes Public
Yes
Lecturer Intro
I studied Applied Mathematics and Physics at the Moscow Institute of Physics and Technology, where I earned both my B.Sc. and M.Sc. degrees. In 2013, I joined the graduate program in Mathematics at Rutgers, The State University of New Jersey, and completed my Ph.D. in 2018 under the guidance of Prof. V. Retakh. After earning my doctorate, I held postdoctoral positions at the University of California Berkeley, the Centre de Recherches Mathématiques in Montreal, and the University of Toronto. In July 2024, I became an Associate Professor at the Beijing Institute of Mathematical Sciences and Applications (BIMSA)