{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "B33WNK1SVN"
    },
    {
      "field": "uses",
      "target": "YHFP9HGBTT"
    }
  ],
  "body": "\n## Summary\n\nA Harish-Chandra module is the algebraic shadow of a representation of a real reductive group $G$.\nIt keeps the vectors that transform in a finite way under a maximal compact subgroup $K$.\nOn them it keeps the actions of $K$ and of the complexified Lie algebra $\\mathfrak g$, and it forgets the topology.\nHarish-Chandra showed that an [[admissible representation|SZV8Y806Y7]] is determined, up to a natural equivalence, by its Harish-Chandra module.\nThis turns questions about infinite-dimensional representations into questions of algebra.\n\n## Explanations\n\n### L1 human\n\nA representation of a group like $\\mathrm{SL}_2(\\R)$ acts on a huge space of functions, where analysis is hard.\nInside it sit the vectors that only move in a finite-dimensional way under rotations, such as trigonometric polynomials on a circle.\nOn these vectors the group no longer acts, but its infinitesimal version, the Lie algebra, still does, together with the rotations.\nThe resulting algebraic object is the Harish-Chandra module.\nFor $\\mathrm{SL}_2(\\R)$ it can be drawn as a row of points, one per frequency $n$, with arrows that raise or lower $n$ by $2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22E47",
      "22E46"
    ],
    "name": "Harish-Chandra module",
    "same_as": {
      "wikidata": "Q4544947",
      "wikipedia": "(g,K)-module"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/(g,K)-module"
      }
    ],
    "tag": "2MKHH636C3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2MKHH636C3/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2MKHH636C3.md",
  "msc": [
    "22E47",
    "22E46"
  ],
  "name": "Harish-Chandra module",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2MK/2MKHH636C3.md",
  "status": null,
  "tag": "2MKHH636C3",
  "uri": "m2://2MKHH636C3",
  "url": "https://tamnd.github.io/math2/wiki/harish-chandra-module/",
  "writing": {
    "tag": "2MKHH636C3",
    "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
      }
    ]
  }
}