On the complete integrability of the periodic quantum Toda lattice

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and $E_6$ by Goodman and Wallach at the beginning of the 1980's. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.