Quantum Cohomology of Flag Manifolds G/B and Quantum Toda Lattices

Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T ) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B. The Toda lattice for (G,B, T ) is the dynamical system on the cotangent bundle T ∗Lie (T ) endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper,