Reflection equations and Littlewood identities for spin Hall-Littlewood and q-Whittaker functions

We describe two-parametric Littlewood identities for inhomogeneous stable spin Hall-Littlewood and inhomogeneous spin q-Whittaker symmetric functions. These identities generalize the Littlewood identity found by Warnaar (2006) for the ordinary Hall-Littlewood polynomials. However, instead of Rogers-Szegö coefficients used by Warnaar, coefficients in our identities are given by products of Al-Salam-Chihara polynomials. Our proof relies on integrable lattice models. A key step includes three new solutions to the reflection and boundary equations for the fully fused six-vertex model. To find these solutions we use isomorphisms between tensor products of evaluation Uq(Lsl₂)-modules, as well as covectors stable under a q-Onsager coideal subalgebra.