On Algebraic Integrability of Gelfand-Zeitlin fields

TL;DR

This paper extends the algebraic integrability of Gelfand-Zeitlin vector fields to all strongly regular elements in complex general linear Lie algebras, using stratification and Poisson geometry techniques.

Contribution

It generalizes previous results by constructing an étale cover and integrating vector fields into a commutative algebraic group action for all strongly regular elements.

Findings

Stratification of strongly regular set by decomposition classes.

Construction of an étale cover that is smooth and irreducible.

Integration of Gelfand-Zeitlin vector fields into a group action.

Abstract

We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to stratify the strongly regular set by subvarieties $X_{D}$. We construct an \'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and $\hat{\mathfrak{g}}$ are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on $X_{D}$ to Hamiltonian vector fields on $\hat{\mathfrak{g}}$ and integrate these vector fields to an action of a connected, commutative algebraic group.