Dualities in the q‐Askey Scheme and degenerate DAHA

The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials 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}). 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.