Zonal Spherical Functions of Quantum Symmetric Pairs

We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder–Macdonald polynomials. To this end, the methods of Letzter’s 2004 paper are translated to modern conventions and right coideal subalgebras, and extended to non-standard cases and cases with non-reduced restricted root systems. For the last elusive Satake type, FII, a conjecture is provided.