关于Gelfand-Zeitlin域的代数可积性
On Algebraic Integrability of Gelfand-Zeitlin fields
摘要：
本文将Kostant和Wallach关于Gelfand-Zeitlin向量场的代数可积性的一个结果推广到$gl(n,\mathbb{C})$中强正则元的全集上.利用分解类将强正则集按子簇$X_{D}$分层。构造了$X_{D}$的一个代数覆盖$\hat{\mathfrak{g}}$，证明了$X_{D}$和$\hat{\mathfrak{g}}$是光滑不可约的.然后，我们使用泊松几何解除$X_{D}$上的Gelfand-Zeitlin向量场到$\hat{\mathfrak{g}}$上的Hamilton向量场，并将这些向量场集成到一个连通的交换代数群的作用。
英文摘要：
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 étale 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.
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