{
  "id": "1507.07410",
  "version": "v1",
  "published": "2015-07-27T14:17:41.000Z",
  "updated": "2015-07-27T14:17:41.000Z",
  "title": "Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings",
  "authors": [
    "Fernando Szechtman"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $A$ be a ring with $1\\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\\leq 2$. Let $H_n(A)$ be the additive group of all $n\\times n$ hermitian matrices over $A$ relative to $*$. Let ${\\mathcal U}_n(A)$ be the subgroup of $\\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\\rtimes {\\mathcal U}_n(A)$, where ${\\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group $\\mathrm{Sp}_{2n}(A)$, if $*=1_A$ (in which case $A$ is commutative), or a unitary group $\\mathrm{U}_{2n}(A)$ if $*\\neq 1_A$. In this paper we construct and classify a family of irreducible representations of $P$ over a field $F$ that is essentially arbitrary. In particular, when $A$ is finite and $F=\\mathbb C$ we obtain irreducible representations of $P$ of the highest possible degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-27T14:17:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irreducible representations",
      "unitary group",
      "unipotent subgroup",
      "upper triangular matrices",
      "main diagonal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150707410S"
    }
  }
}