Kostant-Toda lattices and the universal centralizer

Abstract

To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. These systems are similar in that each exploits and closely reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley, Kostant, and others. One also has Kostant's description of level sets in the Kostant-Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant-Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary results include some qualitative features of the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.