  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 XXZ spin-$\frac{1}{2}$ 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 XXZ 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.
