The Hecke algebras of finite and affine Weyl groups arise naturally as convolution algebras associated to finite and locally compact groups, and play a prominent role in the representation theory of finite groups of Lie type and of reductive p-adic groups. In the early 90’s a class of algebras, known as double affine Hecke algebras, were introduced by Cherednik in connection with affine quantum Knizhnik-Zamolodchikov equations, and subsequently used by him to give uniform proofs of the Macdonald conjectures for root systems. Since then, profound connections have been discovered between double affine Hecke algebras and a broad spectrum of areas in mathematics, including combinatorics, algebraic geometry, number theory (Gauss–Selberg sums), representation theory, harmonic analysis, knot theory, special functions, many body problems, and conformal field theory.

The n-extended affine Lie algebras constitute a class that includes finite dimensional simple Lie algebras (n = 0), irreducible affine Lie algebras (n = 1), and toroidal Lie algebras. For n = 2, closely related Lie algebras appeared in work of Saito and Slodowy in the context of deformation theory of simple elliptic singularities, and in full generality in work of Høegh-Krohn and Torrésani in the context of quantum gauge theory.

Recent results also hint at a new and promising connection between double affine Hecke algebras and Lie superalgebras. Lie superalgebras were initially introduced by physicists in connection with supersymmetry, but the pioneering works of Bernstein, Berezin, Leites and Manin, among others, created a mathematical theory of these algebras. On the other hand, double affine Hecke algebras and their associated Macdonald polynomials can be regarded as a far reaching generalization of the theory of invariant differential operators and spherical polynomials to symmetric spaces. These invariant differential operators, or more precisely their radial parts, admit natural deformations, which also arise in mathematical physics as the completely integrable Calogero–Moser–Sutherland (CMS) systems. The analogous problems for Lie superalgebras and symmetric superspaces have only recently begun to be explored, however there is already a substantial and growing literature on the subject.

There will also be a poster session for junior participants.