{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[unimodular|CXG6JZGMCK]] [[locally compact group|AHV05ZBCYW]] with a [[Haar measure|VY0HXD3GC4]] $dg$.\nLet $\\lambda$ be the left regular representation of $G$ on $L^2(G)$, given by $(\\lambda(g) f)(x) = f(g^{-1} x)$.\nLet $\\pi$ be an [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] of $G$ on a Hilbert space $H$.\nThen $\\pi$ is a *discrete series representation* if it is unitarily equivalent to a subrepresentation of $\\lambda$.\n\n## Remarks\n\nFor $u, v \\in H$, the matrix coefficient of $\\pi$ is the function $g \\mapsto \\ip{\\pi(g) u}{v}$.\nEquivalently, $\\pi$ is in the discrete series if and only if some nonzero matrix coefficient lies in $L^2(G)$.\nThen every matrix coefficient lies in $L^2(G)$.\n\nExample: let $G$ be a compact group, with Haar measure of total mass $1$.\nEach matrix coefficient of a unitary representation is continuous and bounded by $\\norm{u} \\norm{v}$, so it lies in $L^2(G)$.\nSo every irreducible unitary representation of $G$ is in the discrete series.\n\nNon-example: the irreducible unitary representations of $G = \\R$ are the characters $x \\mapsto e^{ixt}$ with $t \\in \\R$.\nEach matrix coefficient has constant absolute value, so it is not square-integrable on $\\R$.\nSo $\\R$ has no discrete series, and $L^2(\\R)$ decomposes as a continuous direct integral by Fourier analysis.\n\nFor groups with non-compact center, such as $\\mathrm{GL}_n(\\R)$, one asks instead that the matrix coefficients be square-integrable modulo the center.\nFor connected semisimple $G$ with finite center and maximal compact subgroup $K$, discrete series exist if and only if $\\rank G = \\rank K$, by a theorem of Harish-Chandra.\nFor example, $\\mathrm{SL}_2(\\R)$ has discrete series, while $\\mathrm{SL}_2(\\C)$ has none.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9AB7HYGPEK",
    "kind": "definition",
    "msc": [
      "22E46",
      "22D10"
    ],
    "name": "Discrete series representation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Discrete_series_representation"
      }
    ],
    "tag": "2CKZETC3QA",
    "uses": [
      "AHV05ZBCYW",
      "CXG6JZGMCK",
      "VY0HXD3GC4",
      "74BPYJNZTM",
      "FX3NEFYFFN",
      "7KGF78FGEF"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2CKZETC3QA/",
  "links": [
    {
      "field": "defines",
      "target": "9AB7HYGPEK"
    },
    {
      "field": "uses",
      "target": "AHV05ZBCYW"
    },
    {
      "field": "uses",
      "target": "CXG6JZGMCK"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "74BPYJNZTM"
    },
    {
      "field": "uses",
      "target": "FX3NEFYFFN"
    },
    {
      "field": "uses",
      "target": "7KGF78FGEF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2CKZETC3QA.md",
  "msc": [
    "22E46",
    "22D10"
  ],
  "name": "Discrete series representation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2CK/2CKZETC3QA.md",
  "status": null,
  "tag": "2CKZETC3QA",
  "uri": "m2://2CKZETC3QA",
  "url": "https://tamnd.github.io/math2/wiki/discrete-series-representation-definition/",
  "writing": {
    "tag": "2CKZETC3QA",
    "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
      }
    ]
  }
}