Hessenberg varieties, Slodowy slices, and integrable systems

TL;DR

This paper explores the deep connections between Hessenberg varieties, Slodowy slices, and integrable systems, extending the Toda lattice framework and revealing new geometric and algebraic structures within complex semisimple groups.

Contribution

It constructs a Poisson variety with an integrable system encompassing Hessenberg varieties and Toda leaves, linking these structures through new geometric insights.

Findings

The total space $X(H_0)$ is a Poisson variety with a complete integrable system.

$X(H_0)$ contains an open dense symplectic leaf isomorphic to $G/Z imes S_{reg}$.

The work connects integrable systems on Hessenberg varieties with those on $G imes S_{reg}$.

Abstract

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group $G$ and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $X(H_0)$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko--Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $X(H_0)$. We also show $X(H_0)$ to have an open dense symplectic leaf isomorphic to $G/Z \times S_{\text{reg}}$, where $Z$ is the centre of $G$ and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. This allows us to invoke results about integrable systems on $G\times S_{\text{reg}}$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.