{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with a [[Haar measure|VY0HXD3GC4]], and let $K$ be a [[compact|0R6EWFKRCF]] [[subgroup|0X1JXT47GA]] of $G$.\nA function $f$ on $G$ is *$K$-bi-invariant* if $f(k_1 g k_2) = f(g)$ for all $g \\in G$ and $k_1, k_2 \\in K$.\nLet $C_c(K \\backslash G / K)$ be the space of continuous, compactly supported, $K$-bi-invariant functions on $G$.\nIt is closed under [[convolution|2Z29JN3VES]].\nThe pair $(G, K)$ is a *Gelfand pair* if $f_1 * f_2 = f_2 * f_1$ for all $f_1, f_2 \\in C_c(K \\backslash G / K)$.\n\n## Remarks\n\nExample: $G = \\mathrm{SL}_2(\\R)$ and $K = \\mathrm{SO}(2)$.\nLet $\\theta(g) = g^{T}$, the transpose, which reverses products and preserves Haar measure.\nBy the Cartan decomposition $g = k_1 a k_2$ with $a$ diagonal, so $g^{T} = k_2^{-1} a k_1^{-1}$ lies in $K g K$.\nSo every bi-invariant $f$ satisfies $f \\circ \\theta = f$, and then $f_1 * f_2 = (f_2 * f_1) \\circ \\theta = f_2 * f_1$.\nThis is Gelfand's trick.\n\nNon-example: $G$ a finite non-abelian group and $K = \\set{e}$.\nThen every function is bi-invariant, and the convolution of the indicators of $\\set{a}$ and $\\set{b}$ is the indicator of $\\set{ab}$.\nSo the algebra is not commutative when $ab \\ne ba$.\n\nEquivalent form: $(G, K)$ is a Gelfand pair exactly when every irreducible unitary representation of $G$ has a space of $K$-fixed vectors of dimension at most $1$.\nFor finite $G$, this says that the permutation representation of $G$ on functions on $G/K$ is multiplicity-free.\nIf $(G, K)$ is a Gelfand pair, then $G$ is unimodular.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "W38ZDXDVYC",
    "kind": "definition",
    "msc": [
      "43A90",
      "22D10",
      "43A85"
    ],
    "name": "Gelfand pair",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gelfand_pair"
      }
    ],
    "tag": "GPQ5M0SA47",
    "uses": [
      "AHV05ZBCYW",
      "VY0HXD3GC4",
      "2Z29JN3VES",
      "0R6EWFKRCF",
      "0X1JXT47GA"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/GPQ5M0SA47/",
  "links": [
    {
      "field": "defines",
      "target": "W38ZDXDVYC"
    },
    {
      "field": "uses",
      "target": "AHV05ZBCYW"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    },
    {
      "field": "uses",
      "target": "2Z29JN3VES"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    },
    {
      "field": "uses",
      "target": "0X1JXT47GA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GPQ5M0SA47.md",
  "msc": [
    "43A90",
    "22D10",
    "43A85"
  ],
  "name": "Gelfand pair",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GPQ/GPQ5M0SA47.md",
  "status": null,
  "tag": "GPQ5M0SA47",
  "uri": "m2://GPQ5M0SA47",
  "url": "https://tamnd.github.io/math2/wiki/gelfand-pair-definition/",
  "writing": {
    "tag": "GPQ5M0SA47",
    "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
      }
    ]
  }
}