{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nDefine the following operator on compactly supported functions on $\\Gamma\\backslash G$:\n\n$$R(\\phi) = \\int_G \\phi(x)R(x)\\,dx.$$\n\nIt extends continuously to $L^2(\\Gamma\\setminus G)$ and for $f\\in L^2(\\Gamma\\setminus G)$ we have:\n\n$$(R(\\phi)f)(x) = \\int_G\\phi(y)f(xy)\\,dy = \\int_{\\Gamma\\setminus G}\\left(\\sum_{\\gamma\\in\\Gamma}\\phi(x^{-1}\\gamma y)\\right)f(y)\\,dy$$\n\nafter a change of variables. Assuming $\\Gamma\\setminus G$ is compact, the operator $R(\\phi)$ is trace-class and the trace formula is the result of computing its trace in two ways as explained below.\n\nThe trace of $R(\\phi)$can be expressed as the integral of the kernel $K(x,y)=\\sum_{\\gamma\\in\\Gamma}\\phi(x^{-1}\\gamma y)$ along the diagonal, that is:\n\n$$\\operatorname{tr}R(\\phi) = \\int_{\\Gamma\\setminus G}\\sum_{\\gamma\\in\\Gamma}\\phi(x^{-1}\\gamma x)\\,dx.$$\n\nLet $\\{\\Gamma\\}$ denote a collection of representatives of conjugacy classes in $\\Gamma$, and $\\Gamma^\\gamma$ and $G^\\gamma$ the respective centralizers of $\\gamma$. Then the above integral can, after manipulation, be written\n\n$$\\operatorname{tr}R(\\phi) = \\sum_{\\gamma\\in\\{\\Gamma\\}} a_\\Gamma^G(\\gamma)\\int_{G^\\gamma\\setminus G}\\phi(x^{-1}\\gamma x)\\,dx.$$\n\nThis gives the *geometric side* of the trace formula.\n\nThe *spectral side* of the trace formula comes from computing the trace of $R(\\phi)$ using the decomposition of the regular representation of $G$ into its irreducible components. Thus\n\n$$\\operatorname{tr}R(\\phi) = \\sum_{\\pi\\in\\hat G}a_\\Gamma^G(\\pi)\\operatorname{tr}\\pi(\\phi)$$\n\nwhere $\\hat G$ is the set of irreducible unitary representations of $G$ (recall that the positive integer $a_\\Gamma^G(\\pi)$ is the multiplicity of $\\pi$ in the unitary representation $R$ on $L^2(\\Gamma\\setminus G)$).\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "kind": "proof",
    "name": "Proof of Selberg trace formula for compact hyperbolic surfaces",
    "proves": "7CHW554E10",
    "sources": [
      {
        "locator": "Selberg trace formula, section The general Selberg trace formula for cocompact quotients \u003e Proof",
        "mention": "wikipedia://en/981694#I01Z",
        "note": "adopted",
        "sections": [
          "Proof"
        ],
        "wikipedia": "en:981694@r1371813961"
      }
    ],
    "tag": "EPTDFQ0HH3"
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/EPTDFQ0HH3/",
  "links": [
    {
      "field": "proves",
      "target": "7CHW554E10"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/EPTDFQ0HH3.md",
  "msc": null,
  "name": "Proof of Selberg trace formula for compact hyperbolic surfaces",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EPT/EPTDFQ0HH3.md",
  "status": null,
  "tag": "EPTDFQ0HH3",
  "uri": "m2://EPTDFQ0HH3",
  "url": "https://math2.tamnd.com/wiki/proof-of-selberg-trace-formula-for-compact-hyperbolic-surfaces/",
  "writing": {
    "tag": "EPTDFQ0HH3",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge uses out",
        "note": "each node that the argument needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}