{
  "id": "0711.4336",
  "version": "v1",
  "published": "2007-11-27T19:56:21.000Z",
  "updated": "2007-11-27T19:56:21.000Z",
  "title": "The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture",
  "authors": [
    "Iain Gordon",
    "Emil Horozov",
    "Milen Yakimov"
  ],
  "comment": "19 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.CO"
  ],
  "abstract": "We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps \\Upsilon \\colon C_n \\to C^n/S_n \\times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture for Grassmannians is equivalent to a special case of our result for Calogero-Moser spaces, namely for the fibres of \\Upsilon over C^n/S_n \\times 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-27T19:56:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational cherednik algebras",
      "real locus",
      "calogero-moser spaces",
      "shapiro conjecture",
      "representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.4336G"
    }
  }
}