{
  "id": "math/0304327",
  "version": "v1",
  "published": "2003-04-22T17:44:01.000Z",
  "updated": "2003-04-22T17:44:01.000Z",
  "title": "Representations of algebraic groups and principal bundles",
  "authors": [
    "Vikram Bhagvandas Mehta"
  ],
  "journal": "Proceedings of the ICM, Beijing 2002, vol. 2, 629--636",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this talk we discuss the relations between representations of algebraic groups and principal bundles on algebraic varieties, especially in characteristic $p$. We quickly review the notions of stable and semistable vector bundles and principal $G$-bundles, where $G$ is any semisimple group. We define the notion of a low height representation in characteristic $p$ and outline a proof of the theorem that a bundle induced from a semistable bundle by a low height representation is again semistable. We include applications of this result to the following questions in characteristic $p$: 1) Existence of the moduli spaces of semistable $G$-bundles on curves. 2) Rationality of the canonical parabolic for nonsemistable principal bundles on curves. 3) Luna's etale slice theorem. We outline an application of a recent result of Hashimoto to study the singularities of the moduli spaces in (1) above, as well as when these spaces specialize correctly from characteristic 0 to characteristic $p$. We also discuss the results of Laszlo-Beauville-Sorger and Kumar-Narasimhan on the Picard group of these spaces. This is combined with the work of Hara and Srinivas-Mehta to show that these moduli spaces are $F$-split for $p$ very large. We conclude by listing some open problems, in particular the problem of refining the bounds on the primes involved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-04-22T17:44:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "14D20"
    ],
    "keywords": [
      "principal bundles",
      "algebraic groups",
      "moduli spaces",
      "low height representation",
      "characteristic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......4327B"
    }
  }
}