{
  "id": "math/0402139",
  "version": "v1",
  "published": "2004-02-09T12:26:07.000Z",
  "updated": "2004-02-09T12:26:07.000Z",
  "title": "Pseudodifferential operators on prehomogeneous vector spaces",
  "authors": [
    "Pablo Ramacher"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G_\\C$ be a connected, linear algebraic group defined over $\\R$, acting regularly on a finite dimensional vector space $V_\\C$ over $\\C$ with $\\R$-structure $V_\\R$. Assume that $V_\\C$ posseses a Zariski-dense orbit, so that $(G_\\C,\\rho,V_\\C)$ becomes a prehomogeneous vector space over $\\R$. We consider the left regular representation $\\pi$ of the group of $\\R$-rational points $G_\\R$ on the Banach space $\\Cvan(V_\\R)$ of continuous functions on $V_\\R$ vanishing at infinity, and study the convolution operators $\\pi(f)$, where $f$ is a rapidly decreasing function on the identity component of $G_\\R$. Denote the complement of the dense orbit by $S_\\C$, and put $S_\\R=S_\\C\\cap V_\\R$. It turns out that the restriction of $\\pi(f)$ to $V_\\R-S_\\R$ is a smooth operator. Furthermore, if $G_\\C$ is reductive, and $S_\\C$ and $S_\\R$ are irreducible hypersurfaces, $\\pi(f)$ corresponds, on each connected component of $V_\\R-S_\\R$, to a totally characteristic pseudodifferential operator. We then investigate the restriction of the Schwartz kernel of $\\pi(f)$ to the diagonal. It defines a distribution on $V_\\R-S_\\R$ given by some power $|p(m)|^s$ of a relative invariant $p(m)$ of $(G_\\C,\\rho,V_\\C)$ and, as a consequence of the fundamental theorem of prehomogeneous vector spaces, its extension to $V_\\R$, and the complex $s$-plane, satisfies functional equations. A trace of $\\pi(f)$ can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-02-09T12:26:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G30",
      "11S90",
      "22E46",
      "47D03"
    ],
    "keywords": [
      "prehomogeneous vector space",
      "finite dimensional vector space",
      "satisfies functional equations",
      "left regular representation",
      "totally characteristic pseudodifferential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......2139R"
    }
  }
}