On q-Deformed gll+1-Whittaker Function III

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