  We construct the action of the quantum group U_v(sl_n) by the natural
correspondences in the equivariant localized $K$-theory of the Laumon based
Quasiflags' moduli spaces. The resulting module is the universal Verma module.
We construct geometrically the Shapovalov scalar product and the Whittaker
vectors. It follows that a certain generating function of the characters of the
global sections of the structure sheaves of the Laumon moduli spaces satisfies
a $v$-difference analogue of the quantum Toda lattice system, reproving the
main theorem of Givental-Lee. The similar constructions are performed for the
affine Lie agebra \hat{sl}_n.
