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 étale 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.
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Partially supported by NSA grant H98230-08-1-0023.
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Colarusso, M., Evens, S. On algebraic integrability of Gelfand-Zeitlin fields. Transformation Groups 15, 46–71 (2010). https://doi.org/10.1007/s00031-010-9082-9
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DOI: https://doi.org/10.1007/s00031-010-9082-9