Spin q-Whittaker polynomials and deformed quantum Toda

Spin \(q\)-Whittaker symmetric polynomials labeled by partitions \(\lambda\) were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable \(\mathfrak{sl}_2\) vertex models. They are a one-parameter deformation of the \(t=0\) Macdonald polynomials. We present a new, more convenient modification of spin \(q\)-Whittaker polynomials and find two Macdonald type \(q\)-difference operators acting diagonally in these polynomials with eigenvalues, respectively, \(q^{-\lambda_1}\) and \(q^{\lambda_N}\) (where \(\lambda\) is the polynomial's label). We study probability measures on interlacing arrays based on spin \(q\)-Whittaker polynomials, and match their observables with known stochastic particle systems such as the \(q\)-Hahn TASEP. In a scaling limit as \(q\nearrow 1\), spin \(q\)-Whittaker polynomials turn into a new one-parameter deformation of the \(\mathfrak{gl}_n\) Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as \(q\nearrow 1\) we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions.