Koornwinder polynomials and the XXZXXZ spin chain

Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey–Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZXXZ spin-12 chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley–Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZXXZ spin chain, in the form of matchmaker representations they relate to Temperley–Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik–Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.