{
  "id": "2002.09969",
  "version": "v1",
  "published": "2020-02-23T18:54:59.000Z",
  "updated": "2020-02-23T18:54:59.000Z",
  "title": "Groups $GL(\\infty)$ over finite fields and multiplications of double cosets",
  "authors": [
    "Yury A. Neretin"
  ],
  "comment": "31pp",
  "categories": [
    "math.RT",
    "math.CT",
    "math.GR"
  ],
  "abstract": "Let $\\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\\mathbb F$, consider the dual space $V^\\diamond$, i. e., the direct product of infinite number of copies of $\\mathbb F$. Consider the direct sum $\\mathbb V=V\\oplus V^\\diamond$ and the group $\\mathbf{GL}$ of all continuous linear operators in $\\mathbb V$. We reduce the theory of unitary representations of $\\mathbf{GL}$ to projective representations of a category whose morphisms are linear relations in finite-dimensional linear spaces over $\\mathbb F$. In fact we consider a certain family $Q_\\alpha$ of subgroups in $\\mathbf{GL}$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_\\alpha$, and reduce this multiplication to products of linear relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-23T18:54:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E66",
      "54H11",
      "18B99",
      "47A06"
    ],
    "keywords": [
      "finite field",
      "double cosets",
      "linear relations",
      "direct sum",
      "infinite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}