Abstract

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 90a€™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 (Gaussa€“Selberg sums), representation theory, harmonic analysis, knot theory, special functions, many body problems, and conformal field theory.