Hecke algebras and harmonic analysis

Abstract

Iwahori–Hecke algebras are ubiquitous. One encounters these algebras in subjects as diverse as harmonic analysis, equivariant K-theory, orthogonal polynomials, quantum groups, knot theory, algebraic combinatorics, and integrable models in statistical physics. In this exposition we will mostly concentrate on the analytic aspects of affine Hecke algebras and study them from the perspective of operator algebras. We will discuss the Plancherel theorem for these type of algebras, and based on a conjectural invariance property of their (operator algebraic) K-theory, study the structure of the tempered dual.