Unitary representations of affine Hecke algebras related to Macdonald spherical functions

TL;DR

This paper constructs a unitary representation of affine Hecke algebras associated with Macdonald spherical functions, revealing their spectral properties and deriving explicit formulas for their structure constants.

Contribution

It introduces a new unitary representation of affine Hecke algebras acting on functions on the weight lattice, connecting to Macdonald spherical functions.

Findings

Center acts diagonally on Macdonald spherical functions

Explicit Pieri formula for Macdonald spherical functions derived

Representation provides new insights into the structure of affine Hecke algebras

Abstract

For any reduced crystallographic root system, we introduce a unitary representation of the (extended) affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight lattice. It is shown that the action of the center under this representation is diagonal on the basis of Macdonald spherical functions. As an application, we compute an explicit Pieri formula for these spherical functions.