{
  "id": "1709.09406",
  "version": "v1",
  "published": "2017-09-27T09:22:50.000Z",
  "updated": "2017-09-27T09:22:50.000Z",
  "title": "Distributions on homogeneous spaces and applications",
  "authors": [
    "N Ressayre"
  ],
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product $odot\\_0$ on thecohomology group $H^*(G/P,{\\mathbb C})$ of any projective $G$-homogeneousspace $G/P$.Their definition uses the notion of Levi-movability for triples ofSchubert varieties in $G/P$.In this article, we introduce a family of $G$-equivariant subbundlesof the tangent bundle of $G/P$ and the associated filtration of the DeRham complex of $G/P$ viewed as a manifold. As a consequence one gets a filtration of the ring $H^*(G/P,{\\mathbb C})$and proves that $\\odot\\_0$ is the associated graded product.One of the aim of this more intrinsic construction of $\\odot\\_0$ isthat there is a natural notion of fundamental class$[Y]\\_{\\odot\\_0}\\in(H^*(G/P),\\odot\\_0)$ for any irreducible subvariety $Y$ of $G/P$.Given two Schubert classes $\\sigma\\_u$ and $\\sigma\\_v$ in$H^*(G/P)$, we define a subvariety $\\Sigma\\_u^v$ of $G/P$. This variety should play the role of the Richardson variety; moreprecisely, we conjecture that$[\\Sigma\\_u^v]\\_{\\odot\\_0}=\\sigma\\_u\\odot\\_0\\sigma\\_v$.We give some evidence for this conjecture, and prove special cases.Finally, we use the subbundles of $TG/P$ to give a geometriccharacterization of the $G$-homogeneous locus of any Schubertsubvariety of $G/P$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-27T09:22:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous spaces",
      "distributions",
      "applications",
      "complex semisimple algebraic group",
      "triples ofschubert varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}