The structure relation for Askey-Wilson polynomials, Zhedanov's algebra AW(3), and a double affine Hecke algebra
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  is implicit in Zhedanov's  algebra AW(3). This will be discussed in the first part of the lecture.
It will turn out  that the algebra AW(3) is closely related to the double affine Hecke algebra (DAHA, ) 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.
 T.H. Koornwinder, The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math., in press; math.CA/0601303.
 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.
 I.G. Macdonald, Affine Hecke algebra and orthogonal polynomials, Cambridge University Press, 2003.
 A.S. Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.