The Macdonald polynomials form a remarkable family of multivariable orthogonal polynomials. They have a wide range of applications in mathematics and mathematical physics, which vary from representation theory of quantum groups and Hecke algebras to integrable systems and conformal field theory.
Important insight into the structure of Macdonald polynomials was provided by Cherednik, who realized the underlying symmetries in terms of explicit difference-reflection operators. His techniques have had (and still have) a tremendous impact on the subject, and has led to proofs of several important conjectures regarding the structure of Macdonald polynomials (the so-called Macdonald conjectures).
This lecture is meant as an introduction to Cherednik's and Macdonald's theory, with emphasis on the special function aspects of the theory. I will avoid the Lie theoretic aspects as much as possible by focussing the attention to an important sub-class of the Macdonald polynomials, known nowadays as the Koornwinder-Macdonald polynomials. I will keep track of the implications and results of the theory in the one-variable setting, in which case the Koornwinder-Macdonald polynomials reduce to the Askey-Wilson polynomials. My lecture will mainly follow the work of Noumi, Sahi and myself on this subject.
If time permits, I will indicate the possible implications of Cherednik's
approach to the Askey-Wilson function transform, which was recently introduced
by Erik Koelink and myself as a generalization of the Jacobi function transform.
The Askey-Wilson function transform will be discussed in more detail in
the first lecture
of Erik Koelink.
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