Weyl modules and q-Whittaker functions
  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. 
Alexander Braverman and Michael Finkelberg
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