Representation theoretic interpretation and interpolation properties of inhomogeneous spin $q$-Whittaker polynomials

We establish new properties of inhomogeneous spin $q$-Whittaker polynomials, which are symmetric polynomials generalizing $t=0$ Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an $R$-matrix, as is often the case, but from other intertwining operators of $U_q^{\mathrm{\prime }}({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$-modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin $q$-Whittaker polynomials in full generality. Moreover, we are able to characterize spin $q$-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of $q$-Whittaker and elementary symmetric polynomials.