{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a connected noncompact simple real Lie group, contained in its simply connected [[complexification|DRYYK2WX0M]] $G_\\C$.\nLet $K \\subset G$ be a maximal compact subgroup such that $X = G/K$ is a [[Hermitian symmetric space|NT4797Z9H8]].\nLet $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}$ be the Cartan decomposition, and let $\\mathfrak{p}_\\C = \\mathfrak{p}^+ \\oplus \\mathfrak{p}^-$ be the splitting into the $\\pm i$ eigenspaces of the complex structure.\nPut $P^\\pm = \\exp \\mathfrak{p}^\\pm$, and let $K_\\C$ be the complexification of $K$ in $G_\\C$.\nThen:\n\n1. Every $g \\in G$ can be written uniquely as $g = p^+ k p^-$ with $p^\\pm \\in P^\\pm$ and $k \\in K_\\C$.\n2. The map $\\zeta \\colon X \\to \\mathfrak{p}^+$, $gK \\mapsto \\log p^+$, is a holomorphic embedding onto a bounded domain in the complex vector space $\\mathfrak{p}^+ \\cong \\C^n$, where $n = \\dim_\\C X$.\n\n## Remarks\n\nThe image is a [[bounded symmetric domain|Y7Q2SNKG0C]], the Harish-Chandra realization of $X$, and it is convex and circled.\nThe map extends to the Borel embedding of $X$ as an open subset of the compact dual $G_\\C / K_\\C P^-$.\nIn this picture $\\zeta$ is the restriction of an affine chart $P^+ \\cong \\mathfrak{p}^+$ of the compact dual.\nThe same holds for semisimple $G$ by taking products.\n\nExample: for $G = SU(1, 1)$ and $K = U(1)$ the Harish-Chandra realization of $G/K$ is the unit disc.\nFor $G = Sp(2n, \\R)$ and $K = U(n)$, the space $G/K$ is the [[Siegel upper half-space|5PTYE97PF6]] $\\mathbb{H}_n$, and the realization is the domain $\\set{W : W^T = W,\\ I - W^* W \u003e 0}$, reached by the Cayley transform $Z \\mapsto (Z - iI)(Z + iI)^{-1}$.\n\n## Explanations\n\n### L1 human\n\nA Hermitian symmetric space of noncompact type is first defined abstractly, as a quotient of a Lie group.\nThe theorem says that it can always be drawn as a bounded region in $\\C^n$.\nSo abstract symmetric spaces and concrete bounded symmetric domains are the same objects.\n\n### L2 human\n\nTogether with the fact that bounded symmetric domains with the Bergman metric are Hermitian symmetric, this gives an equivalence between the two notions.\nThe embedding is the tool behind holomorphic discrete series representations and the analysis of automorphic forms on $G/K$.\nIt also explains why $X$ sits as an open subset inside its compact dual, as the disc sits inside the Riemann sphere.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "NT4797Z9H8",
      "Y7Q2SNKG0C"
    ],
    "kind": "statement",
    "msc": [
      "32M15",
      "22E46"
    ],
    "name": "Harish-Chandra embedding",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hermitian_symmetric_space"
      }
    ],
    "tag": "4RXXE3BB6W",
    "uses": [
      "NT4797Z9H8",
      "Y7Q2SNKG0C",
      "7JBRGCVJRB",
      "DRYYK2WX0M"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/4RXXE3BB6W/",
  "links": [
    {
      "field": "about",
      "target": "NT4797Z9H8"
    },
    {
      "field": "about",
      "target": "Y7Q2SNKG0C"
    },
    {
      "field": "uses",
      "target": "NT4797Z9H8"
    },
    {
      "field": "uses",
      "target": "Y7Q2SNKG0C"
    },
    {
      "field": "uses",
      "target": "7JBRGCVJRB"
    },
    {
      "field": "uses",
      "target": "DRYYK2WX0M"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4RXXE3BB6W.md",
  "msc": [
    "32M15",
    "22E46"
  ],
  "name": "Harish-Chandra embedding",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4RX/4RXXE3BB6W.md",
  "status": null,
  "tag": "4RXXE3BB6W",
  "uri": "m2://4RXXE3BB6W",
  "url": "https://tamnd.github.io/math2/wiki/harish-chandra-embedding/",
  "writing": {
    "tag": "4RXXE3BB6W",
    "kind": "statement",
    "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
      }
    ]
  }
}