{
  "id": "1510.07274",
  "version": "v1",
  "published": "2015-10-25T17:35:30.000Z",
  "updated": "2015-10-25T17:35:30.000Z",
  "title": "A uniform classification of discrete series representations of affine Hecke algebras",
  "authors": [
    "Dan Ciubotaru",
    "Eric Opdam"
  ],
  "comment": "31 pages, 2 tables",
  "categories": [
    "math.RT"
  ],
  "abstract": "We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\\mathcal{H}_{\\mathbf{v}}$, in terms of a canonically defined basis $\\mathcal{B}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\\mathcal{H}$, and to all $\\mathbf{v}\\in\\mathcal{Q}$, where $\\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\\mathcal{H}$. By analytic Dirac induction we define for each $b\\in \\mathcal{B}_{gm}$ a continuous (in the sense of [OS2]) family $\\mathcal{Q}^{reg}_b:=\\mathcal{Q}_b\\backslash\\mathcal{Q}_b^{sing}\\ni\\mathbf{v}\\to\\operatorname{Ind}_{D}(b;\\mathbf{v})$, such that $\\epsilon(b;\\mathbf{v})\\operatorname{Ind}_{D}(b;\\mathbf{v})$ (for some $\\epsilon(b;\\mathbf{v})\\in\\{\\pm 1\\}$) is an irreducible discrete series character of $\\mathcal{H}_{\\mathbf{v}}$. Here $\\mathcal{Q}^{sing}_b\\subset\\mathcal{Q}$ is a finite union of hyperplanes in $\\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\\operatorname{Ind}_{D}(b;\\mathbf{v})$ are piecewise rational in the parameters $\\mathbf{v}$. Remarkably, the formal degree of $\\operatorname{Ind}_{D}(b;\\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\\in \\mathcal{B}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\\epsilon(b;\\mathbf{v})\\operatorname{Ind}_{D}(b;\\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\\epsilon(b;\\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-25T17:35:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "22D25",
      "43A30"
    ],
    "keywords": [
      "discrete series representations",
      "uniform classification",
      "formal degree",
      "virtual discrete series characters",
      "semisimple affine hecke algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}