{
  "id": "1508.01664",
  "version": "v1",
  "published": "2015-08-07T12:01:45.000Z",
  "updated": "2015-08-07T12:01:45.000Z",
  "title": "Higher symmetries of powers of the Laplacian and rings of differential operators",
  "authors": [
    "T. Levasseur",
    "J. T. Stafford"
  ],
  "categories": [
    "math.RT",
    "math.DG"
  ],
  "abstract": "We study the interplay between the minimal representations of the orthogonal Lie algebra $\\mathfrak{g}=\\mathfrak{so}(n+2,\\mathbb{C})$ and the \\emph{algebra of symmetries} $\\mathscr{S}(\\Box^r)$ of powers of the Laplacian $\\Box$ on $\\mathbb{C}^{n}$. The connection is made through the construction of highest weight representation of $\\mathfrak{g}$ via the ring of differential operators $\\mathcal{D}(X)$ on the singular scheme $X=(F^r=0)\\subset \\mathbb{C}^n$, where $F$ is the sum of squares. In particular we prove that $ \\mathscr{S}(\\Box^r)\\cong \\mathcal{D}(X)$ is isomorphic to a primitive factor ring of $U(\\mathfrak{g})$. Interestingly, if (and only if) $n$ is even with $2r\\geq n$ then both $\\mathcal{D}(X)$ and its natural module $\\mathcal{O}(X)$ have a finite dimensional factor. These results all have real analogues, with $\\Box$ replaced by the d'Alembertian on the pseudo-Euclidean space $\\mathbb{R}^{p,q}$ and $\\mathfrak{g}$ replaced by the real Lie algebra $\\mathfrak{so}(p+1,q+1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-07T12:01:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S32",
      "58J70",
      "17B08"
    ],
    "keywords": [
      "differential operators",
      "higher symmetries",
      "real lie algebra",
      "orthogonal lie algebra",
      "highest weight representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150801664L"
    }
  }
}