{
  "id": "1111.2868",
  "version": "v1",
  "published": "2011-11-11T21:15:55.000Z",
  "updated": "2011-11-11T21:15:55.000Z",
  "title": "The Gelfand-Zeitlin integrable system and K-orbits on the flag variety",
  "authors": [
    "Mark Colarusso",
    "Sam Evens"
  ],
  "comment": "33 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of $n\\times n$ complex matrices $\\fgl(n,\\C)$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of $K_{n}=GL(n-1,\\C)\\times GL(1,\\C)$-orbits on the flag variety $\\mathcal{B}_{n}$ of $GL(n,\\C)$ to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of $K_{n}$ and $GL(n,\\C)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-11T21:15:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flag variety",
      "gelfand-zeitlin integrable system",
      "strongly regular elements",
      "general theory",
      "specific example"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2868C"
    }
  }
}