北京雁栖湖应用数学研究院

we will define and study Coxeter groups and their action by reflections. The main examples of them are Weyl groups of Lie algebras. In the case of finite-dimensional Lie algebras they have a natural extensions called affine Weyl groups. We will focus on the affine analogues of affine Weyl groups. They are called double affine Weyl groups.
Group algebras of Coxeter groups have a natural deformation called Hecke algebras. We will study double affine Hecke algebras which correspond to double affine Weyl groups. These algebras have the most important representations called polynomial representations. These representations have a natural basis called Nonsymetric Macdonald polynomials. We will explain the various properties of these polynomials, relations to representation theory and integrable systems.