{
  "id": "1209.5653",
  "version": "v3",
  "published": "2012-09-25T15:51:09.000Z",
  "updated": "2013-05-05T22:25:47.000Z",
  "title": "Representations with Small $K$ Types",
  "authors": [
    "Seung Won Lee"
  ],
  "comment": "37 pages. Analogous results for the connected, simply connected, split real forms of doubly laced, simple Lie types other than type C_n are added to the results for the connected, simply connected, split real forms of simply laced, simple Lie types of rank \\geq 2 in the previous version",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}_{\\mathbb{R}}$ be a split real, simple Lie algebra with complexification $\\mathfrak{g}$. Let $G_{\\mathbb{C}}$ be the connected, simply connected Lie group with Lie algebra $\\mathfrak{g}$, $G_{\\mathbb{R}}$ the connected subgroup of $G_{\\mathbb{C}}$ with Lie algebra $\\mathfrak{g}_{\\mathbb{R}}$, and $G$ a covering group of $G_{\\mathbb{R}}$ with a maximal compact subgroup $K$. A complete classification of \"small\" $K$ types is derived via Clifford algebras, and an analog, $P^{\\xi}$, of Kostant's $P^{\\gamma}$ matrix is defined for a $K$ type ${\\xi}$ of principal series admitting a small $K$ type. For the connected, simply connected, split real forms of simple Lie types other than type $C_n$, a product formula for the determinant of $P^{\\xi}$ over the rank one subgroups corresponding to the positive roots is proved. We use these results to determine cyclicity of a small $K$ type of principal series in the closed Langlands chamber and irreducibility of the unitary principal series admitting a small $K$ type.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-05T22:25:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46"
    ],
    "keywords": [
      "representations",
      "principal series admitting",
      "maximal compact subgroup",
      "simple lie types",
      "split real forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}