{
  "id": "1710.03185",
  "version": "v1",
  "published": "2017-10-09T16:41:28.000Z",
  "updated": "2017-10-09T16:41:28.000Z",
  "title": "Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials",
  "authors": [
    "Daniel Bump",
    "Maki Nakasuji"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "A problem in representation theory of $p$-adic groups is the computation of the \\textit{Casselman basis} of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-09T16:41:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "20F55",
      "05E15"
    ],
    "keywords": [
      "casselmans basis",
      "kazhdan-lusztig polynomials",
      "iwahori vectors",
      "transition matrix",
      "spherical principal series representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}