Finite difference quantum Toda lattice via equivariant K-theory

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 and Lee (cf.[7]). Similar constructions are performed for the affine Lie algebra sl_n.