{
  "id": "1712.03995",
  "version": "v1",
  "published": "2017-12-11T19:33:12.000Z",
  "updated": "2017-12-11T19:33:12.000Z",
  "title": "A new proof of Harish-Chandra's integral formula",
  "authors": [
    "Colin McSwiggen"
  ],
  "comment": "12 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "We present a new proof of Harish-Chandra's formula $$\\Pi(h_1) \\Pi(h_2) \\int_G e^{\\langle \\mathrm{Ad}_g h_1, h_2 \\rangle} dg = \\frac{ [ \\! [ \\Pi, \\Pi ] \\!] }{|W|} \\sum_{w \\in W} \\epsilon(w) e^{\\langle w(h_1),h_2 \\rangle},$$ where $G$ is a compact, connected, semisimple Lie group, $dg$ is normalized Haar measure, $h_1$ and $h_2$ lie in a Cartan subalgebra of the complexified Lie algebra, $\\Pi$ is the discriminant, $\\langle \\cdot, \\cdot \\rangle$ is the Killing form, $[ \\! [ \\cdot, \\cdot ] \\!]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $\\epsilon(w)$ is the sign of $w \\in W$. The proof in this paper follows from a relationship between heat flow on a semisimple Lie algebra and heat flow on a Cartan subalgebra, extending methods developed by Itzykson and Zuber for the case of an integral over the unitary group $U(N)$. The heat-flow proof allows a systematic approach to studying the asymptotics of orbital integrals over a wide class of groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-11T19:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "15B52"
    ],
    "keywords": [
      "harish-chandras integral formula",
      "heat flow",
      "cartan subalgebra",
      "semisimple lie algebra",
      "killing form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}