On algebraic integrability of Gelfand-Zeitlin fields

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 \( \mathfrak{g}\mathfrak{l} \)(n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties \( {X_\mathcal{D}} \). We construct an etale cover \( {\hat{\mathfrak{g}}}_\mathcal{D} \) of \( {X_\mathcal{D}} \) and show that \( {X_\mathcal{D}} \) and \( {\hat{\mathfrak{g}}}_\mathcal{D} \) are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on \( {X_\mathcal{D}} \) to Hamiltonian vector fields on \( {\hat{\mathfrak{g}}}_\mathcal{D} \) and integrate these vector fields to an action of a connected, commutative algebraic group.