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

Abstract

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