The structure relation for Askey-Wilson polynomials, Zhedanov's algebra AW(3), and a double affine Hecke algebra

Abstract:
Several authors have studied raising, lowwering and structure relations for families of orthogonal polynomials in the Askey and q-Askey scheme. The approaches are usually on a case-by-case basis, and quite computational. However, a quick and more universal approach [1] is implicit in Zhedanov's [4] algebra AW(3). This will be discussed in the first part of the lecture.

It will turn out [2] that the algebra AW(3) is closely related to the double affine Hecke algebra (DAHA, [3]) for the Askey-Wilson polynomials. This will be the main topic of the lecture. To be a little more specific, consider AW(3) with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. This representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the DAHA related to symmetric and non-symmetric Askey-Wilson polynomials will be presented without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) will be introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

[1] T.H. Koornwinder, The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math., in press; math.CA/0601303.
[2] T.H. Koornwinder, The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063; math.QA/0612730.
[3] I.G. Macdonald, Affine Hecke algebra and orthogonal polynomials, Cambridge University Press, 2003.
[4] A.S. Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.