{
  "id": "2307.03857",
  "version": "v1",
  "published": "2023-07-07T22:32:56.000Z",
  "updated": "2023-07-07T22:32:56.000Z",
  "title": "Intermediate Jacobi polynomials for the root system of type BC1",
  "authors": [
    "Max van Horssen",
    "Maarten van Pruijssen"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Intermediate Jacobi polynomials for a root system $R$ with Weyl group $W$ are orthogonal polynomials that are invariant under a parabolic subgroup of $W$. The extreme cases are the symmetric and the non-symmetric Jacobi polynomials studied by Heckman and Opdam. Intermediate Jacobi polynomials can also be understood as vector-valued orthogonal polynomials. We study the rank one case in this paper. The interpretation of the non-symmetric Jacobi polynomials as vector-valued polynomials has interesting consequences. The first is that the non-symmetric Jacobi polynomials can be expressed in terms of the symmetric ones. The second is that we recover a shift operator for the symmetric Jacobi polynomials that comes from the Hecke algebra representation. Thirdly, for geometric root multiplicities the vector-valued polynomials can be identified with spherical functions on the sphere $S^{2n}$ associated to the fundamental spin-representation. In this way the Cherednik-operator appears as a Dirac operator for the spinors on this space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-07T22:32:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C52",
      "33C45",
      "33E30"
    ],
    "keywords": [
      "intermediate jacobi polynomials",
      "root system",
      "non-symmetric jacobi polynomials",
      "type bc1",
      "vector-valued polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}