{
  "id": "2405.01897",
  "version": "v1",
  "published": "2024-05-03T07:40:31.000Z",
  "updated": "2024-05-03T07:40:31.000Z",
  "title": "Orbits and invariants for coisotropy representations",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "23 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "For a subgroup $H$ of a reductive group $G$, let $\\mathfrak m\\subset \\mathfrak g^*$ be the cotangent space of $eH\\in G/H$. The linear action $(H:\\mathfrak m)$ is the coisotropy representation. It is known that the complexity and rank of $G/H$ (denoted $c$ and $r$, respectively) are encoded in properties of $(H:\\mathfrak m)$. We complement existing results on $c$, $r$, and $(H:\\mathfrak m)$, especially for quasiaffine varieties $G/H$. If the algebra of invariants $k[\\mathfrak m]^H$ is finitely generated, then we establish a connection between the nullcones in $\\mathfrak m$ and $\\mathfrak g^*$. Two other topics considered are (i) a relationship between varieties $G/H$ of complexity at most 1 and the homological dimension of the algebra of invariants $k[\\mathfrak m]^H$ and (ii) the Poisson structure of $k[\\mathfrak m]^H$ and Poisson-commutative subalgebras in $k[\\mathfrak m]^H$ with maximal transcendence degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T07:40:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "14R20",
      "14M27",
      "17B20"
    ],
    "keywords": [
      "coisotropy representation",
      "invariants",
      "maximal transcendence degree",
      "complement existing results",
      "cotangent space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}