{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "61Z8MZ2M28"
    }
  ],
  "body": "\n## Summary\n\nLet $K$ be a compact open subgroup of a [[locally compact group|AHV05ZBCYW]] $G$.\nThe Hecke algebra of $(G, K)$ is the algebra of compactly supported $K$-bi-invariant functions on $G$ under [[convolution|2Z29JN3VES]].\nIt has a basis given by the indicator functions of the double cosets $K g K$.\nThe $K$-fixed vectors of a smooth representation of $G$ form a module over it.\nThis makes it a main tool for representations of $p$-adic groups.\nFor $G = \\mathrm{GL}_n(\\Q_p)$ and $K = \\mathrm{GL}_n(\\Z_p)$ it is commutative, and it is closely related to the [[Hecke operators|S4SV3QZ8B1]] on modular forms.\n\n## Explanations\n\n### L1 human\n\nPick a compact open subgroup $K$, like $\\mathrm{GL}_2(\\Z_p)$ inside $\\mathrm{GL}_2(\\Q_p)$.\nThe group is then cut into double cosets $KgK$, which are like coarse blocks of the group.\nThe Hecke algebra records how these blocks multiply: the product of two blocks spreads over several blocks with integer multiplicities.\nFor a discrete group with $K$ trivial, the blocks are single elements and one gets the ordinary group algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22D20",
      "20C08",
      "22E50"
    ],
    "name": "Hecke algebra of a locally compact group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hecke_algebra_of_a_pair"
      }
    ],
    "tag": "6H3WCCHZB5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/6H3WCCHZB5/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/6H3WCCHZB5.md",
  "msc": [
    "22D20",
    "20C08",
    "22E50"
  ],
  "name": "Hecke algebra of a locally compact group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6H3/6H3WCCHZB5.md",
  "status": null,
  "tag": "6H3WCCHZB5",
  "uri": "m2://6H3WCCHZB5",
  "url": "https://tamnd.github.io/math2/wiki/hecke-algebra-of-a-locally-compact-group/",
  "writing": {
    "tag": "6H3WCCHZB5",
    "kind": "concept",
    "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
      }
    ]
  }
}