Slodowy slices and the complete integrability of Mishchenko Fomenko subalgebras on regular adjoint orbits
This work is concerned with Mishchenko Fomenko subalgebras and their restrictions to the adjoint orbits in a finite dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve upon this result, showing that each Mishchenko Fomenko subalgebra yields a completely integrable system on all regular orbits (i.e. orbits of maximal dimension) Our approach incorporates the theory of regular \mathfrak{sl}_2$ triples and associated Slodowy slices, as developed by Kostant.