{
  "id": "1909.01096",
  "version": "v1",
  "published": "2019-09-03T11:51:42.000Z",
  "updated": "2019-09-03T11:51:42.000Z",
  "title": "Principal Series Representation of $SU(2,1)$ and Its Intertwining Operator",
  "authors": [
    "Zhuohui Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, following a similar procedure developed by Buttcane and Miller in \\cite{MillerButtcane} for $SL(3,\\RR)$, the $(\\frakg,K)$-module structure of the minimal principal series of real reductive Lie groups $SU(2,1)$ is described explicitly by realizing the representations in the space of $K$-finite functions on $U(2)$. Moreover, by combining combinatorial techniques and contour integrations, this paper introduces a method of calculating intertwining operators on the principal series. Upon restriction to each $K$-type, the matrix entries of intertwining operators are represented by $\\Gamma$-functions and Laurent series coefficients of hypergeometric series. The calculation of the $(\\frakg,K)$-module structure of principal series can be generalized to real reductive Lie groups whose maximal compact subgroup is a product of $SU(2)$'s and $U(1)$'s.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-03T11:51:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "11F55",
      "33C47"
    ],
    "keywords": [
      "principal series representation",
      "intertwining operator",
      "real reductive lie groups",
      "module structure",
      "maximal compact subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}