Slodowy slices and the complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits

TL;DR

This paper proves that Mishchenko-Fomenko subalgebras generate completely integrable systems on all regular adjoint orbits in complex semisimple Lie algebras, extending previous results to a broader class of orbits.

Contribution

It demonstrates that Mishchenko-Fomenko subalgebras induce integrable systems on all regular orbits, not just those in general position, using Slodowy slices and $rak{sl}_2$-triples.

Findings

Complete integrability on all regular orbits.

Extension of previous results beyond general position.

Use of Slodowy slices and Kostant's theory.

Abstract

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.