On algebraic integrability of Gelfand-Zeitlin fields

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 mathfrakgmathfrakl mathfrak{g}mathfrak{l} (n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties X_D {X_mathcal{D}} . We construct an étale cover [^(mathfrakg)]_D {hat{mathfrak{g}}}_mathcal{D} of X_D {X_mathcal{D}} and show that X_D {X_mathcal{D}} and [^(mathfrakg)]_D {hat{mathfrak{g}}}_mathcal{D} are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on X_D {X_mathcal{D}} to Hamiltonian vector fields on [^(mathfrakg)]_D {hat{mathfrak{g}}}_mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group.