{
  "id": "0912.5132",
  "version": "v4",
  "published": "2009-12-28T05:31:15.000Z",
  "updated": "2011-12-13T21:46:58.000Z",
  "title": "Affine Gindikin-Karpelevich formula via Uhlenbeck spaces",
  "authors": [
    "Alexander Braverman",
    "Michael Finkelberg",
    "David Kazhdan"
  ],
  "comment": "A few misprints are corrected; some references added",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over a local field of positive characteristic. The proof is geometric and it is based on the results of [1] about intersection cohomology of certain Uhlenbeck-type moduli spaces (in fact, our proof is conditioned upon the assumption that the results of [1] are valid in positive characteristic). In particular, we give a geometric explanation of certain combinatorial differences between finite-dimensional and affine case (observed earlier by Macdonald and Cherednik), which here manifest themselves by the fact that the affine Gindikin-Karpelevich formula has an additional term compared to the finite-dimensional case. Very roughy speaking, that additional term is related to the fact that the loop group of an affine Kac-Moody group (which roughly speaking should be thought of as some kind of \"double loop group\") does not behave well from algebro-geometric point of view; however it has a better behaved version which has something to do with algebraic surfaces. A uniform (i.e. valid for all local fields) and unconditional (but not geometric) proof of the affine Gindikin-Karpelevich formula is going to appear in [2].",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-12-13T21:46:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine gindikin-karpelevich formula",
      "uhlenbeck spaces",
      "local field",
      "additional term",
      "loop group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.5132B"
    }
  }
}