Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

An infinite-dimensional representation of the double affine Hecke algebra of rank 1 and type \((C_1^{\vee },C_1)\) in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey–Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.