  Let G be a semi-simple simply connected group over complex numbers. In this
paper we give a geometric definition of the (dual) Weyl modules over the group
G[t] and show that their characters form an eigen-function of the lattice
version of the q-Toda integrable integrable system (defined by means of the
quantum group version of Kostant-Whittaker reduction due to Etingof and
Sevostyanov). All the proofs are algebro-geometric and rely on our previous
work which interprets the universal eigen-function of the q-Toda system in
terms of rings of functions on the spaces of based quasi-maps from P^1 to the
flag variety of G.
  We discuss in detail the relation between the current work and the works of
Cherednik, Ion, Sanderson and Gerasimov-Lebedev-Oblezin.
