{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[real reductive Lie group|42ZCE14AD2]] with a [[Haar measure|VY0HXD3GC4]] $dg$.\nLet $\\pi$ be an [[admissible representation|SZV8Y806Y7]] of $G$ of finite length on a Hilbert space $H$.\nFor a [[smooth function|0V33BNKZ9S]] $f \\colon G \\to \\C$ with compact support, let\n$$\\pi(f) = \\int_G f(g) \\, \\pi(g) \\, dg,$$\na bounded operator on $H$.\nHarish-Chandra proved that each $\\pi(f)$ is a [[trace class operator|Z2CP530WDE]].\nThe *Harish-Chandra character* of $\\pi$ is the distribution\n$$\\Theta_\\pi(f) = \\operatorname{tr} \\pi(f), \\qquad f \\in C_c^\\infty(G).$$\n\n## Remarks\n\nExample: if $\\dim H \u003c \\infty$, then $\\operatorname{tr} \\pi(f) = \\int_G f(g) \\operatorname{tr} \\pi(g) \\, dg$ by linearity of the trace.\nSo $\\Theta_\\pi$ is the distribution given by the ordinary [[character|RMWMYBY8TC]] $g \\mapsto \\operatorname{tr} \\pi(g)$.\n\nNon-example: for $\\dim H = \\infty$, the operator $\\pi(e) = \\id_H$ is not trace class.\nSo the naive rule $g \\mapsto \\operatorname{tr} \\pi(g)$ does not define a function, and the smearing by $f$ is necessary.\n\nThe character $\\Theta_\\pi$ is invariant under conjugation: $\\Theta_\\pi(f^x) = \\Theta_\\pi(f)$ for $f^x(g) = f(x g x^{-1})$.\nTwo irreducible admissible representations have the same character if and only if they are infinitesimally equivalent.\nBy [[Harish-Chandra's regularity theorem|M9X7Y3GZZD]], $\\Theta_\\pi$ is given by a locally integrable function.\nThe same definition works for $G(F)$ with $F$ a non-archimedean local field, with $f$ locally constant of compact support.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "F2E3317P78",
    "kind": "definition",
    "msc": [
      "22E46",
      "22E45",
      "22E50"
    ],
    "name": "Harish-Chandra character",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harish-Chandra_character"
      }
    ],
    "tag": "FHKGR7MD1A",
    "uses": [
      "SZV8Y806Y7",
      "42ZCE14AD2",
      "VY0HXD3GC4",
      "Z2CP530WDE",
      "RMWMYBY8TC",
      "0V33BNKZ9S"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FHKGR7MD1A/",
  "links": [
    {
      "field": "defines",
      "target": "F2E3317P78"
    },
    {
      "field": "uses",
      "target": "SZV8Y806Y7"
    },
    {
      "field": "uses",
      "target": "42ZCE14AD2"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "Z2CP530WDE"
    },
    {
      "field": "uses",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "0V33BNKZ9S"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FHKGR7MD1A.md",
  "msc": [
    "22E46",
    "22E45",
    "22E50"
  ],
  "name": "Harish-Chandra character",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FHK/FHKGR7MD1A.md",
  "status": null,
  "tag": "FHKGR7MD1A",
  "uri": "m2://FHKGR7MD1A",
  "url": "https://tamnd.github.io/math2/wiki/harish-chandra-character-definition/",
  "writing": {
    "tag": "FHKGR7MD1A",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}