Double affine hecke algebras in representation theory, combinatorics, geometry, and mathematical physics

Lectures: 2 sessions / week, 1.5 hours / session

Double affine Hecke algebras, also called Cherednik algebras, were introduced by Cherednik in 1993 as a tool in his proof of Macdonald's conjectures about orthogonal polynomials for root systems. Since then, it has been realized that Cherednik algebras are of great independent interest; they appeared in many different mathematical contexts and found many applications.

The goal of this course is to give an introduction to Cherednik algebras, and to review the web of connections between them and other mathematical objects. For this reason, the course consists of several parts that are relatively independent of each other. Also, because of such a wide scope, many important topics are skipped and many proofs are omitted or sketched; we explain in detail only those proofs which can be understood in class and are instructive. We also try to focus on explicit examples.

Course grade is based upon class attendance and participation. There are no homework assignments, projects, or exams.

The rational Cherednik algebra II

Finite Coxeter groups and the Macdonald-Mehta integral

The Knizknik-Zamolodchikov functor

Rational Cherednik algebras for varieties with group actions

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