Whittaker Functions on Quantum Groups and Q-deformed Toda Operators

Let G be a simply connected simple Lie group over C. Let N± be the positive and the negative maximal unipotent subgroups, and H the maximal torus, corresponding to some polarization of G. Let G0 = N−HN+ be the big Bruhat cell. Let χ± : N± → C be holomorphic nondegenerate characters (i.e. they don’t vanish on simple roots). A Whittaker function on G0 with characters χ+, χ− is any holomorphic function φ on G0 such that φ(a−a0a+) = χ−(a−)χ+(a+)φ(a0), where a± ∈ N±, a0 ∈ G0. Thus, a Whittaker function is completely determined by its values on the maximal torus H. In the late 1970’s it was observed (by Kazhdan and Kostant) that the restriction of the Laplace operator on G to Whittaker functions is the quantum Toda Hamiltonian. This allows one to easily prove Kostant’s integrability theorem for the quantum Toda system: quantum integrals are restrictions to Whittaker functions of higher Casimirs of G. This procedure can be generalized to the case when the group G is replaced with the corresponding affine Kac-Moody group Ĝ. In this case, one should consider Whittaker functions “of critical level”, i.e. functions satisfying the equation φ(xz) = φ(x)z ∨ , where z is a central element of Ĝ and h the dual Coxeter number of G. The restriction of the Laplace operator to Whittaker functions is then the quantum affine Toda Hamiltonian. As a result, one gets a proof of the integrability of the quantum affine Toda system: quantum integrals are restrictions to Whittaker functions of higher Casimirs of Ĝ at the critical level defined in [FF,GW]. The goal of this paper is to generalize this theory (both for G and Ĝ) to the case of quantum groups, and give the corresponding proofs of the quantum integrability of the q-difference analogs of the quantum Toda systems for G and Ĝ.