On q-Deformed $${{\mathfrak{gl}}_{\ell+1}}$$-Whittaker Function III

In this paper, we identify q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions with a specialization of the Macdonald polynomials. This provides a representation of q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions in terms of the Demazure characters of affine Lie algebra ^(\mathfrakgl)]_l+1{\widehat{\mathfrak{gl}}_{\ell+1}}. We also define a system of dual Hamiltonians for q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Toda chains and give a new integral representation for the q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions. Finally, we represent the q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker function as a matrix element of a quantum torus algebra.