Dualities in the q-Askey Scheme and Degenerate DAHA

The Askey–Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials Rn[z] that are eigenfunctions of a second-order q-difference operator L, and of a second-order difference operator in the variable n with eigenvalue z + z−1 = 2x. Then, L and multiplication by z + z−1 generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.