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  "body": "\n## Definition\n\nLet $G$ be a connected complex [[semisimple Lie group|XJS07AJ50Y]].\nLet $B$ be a [[Borel subgroup|5JSYA1JNCC]] of $G$, a maximal connected solvable closed complex subgroup.\nLet $P$ be a closed complex subgroup of $G$ with $B \\subseteq P$, a parabolic subgroup.\nThe *flag manifold* of $G$ and $P$ is the [[homogeneous space|85TWR6E75P]] $G/P$, a complex manifold with the holomorphic action of $G$ by left translation.\nWhen $P = B$, the space $G/B$ is the *complete* (or full) flag manifold of $G$.\n\n## Remarks\n\nExample: $G = \\mathrm{SL}_n(\\C)$ with $B$ the upper triangular matrices of determinant $1$.\nThen $G$ acts transitively on complete flags $V_1 \\subset \\cdots \\subset V_{n-1}$ in $\\C^n$ with $\\dim V_i = i$, and $B$ is the stabilizer of the standard flag $V_i = \\operatorname{span}(e_1, \\dots, e_i)$.\nSo $G/B$ is the space of complete flags, of complex dimension $n(n-1)/2$.\nThe subgroup $P$ of matrices with first column in $\\C e_1$ contains $B$, and $G/P$ is the projective space $\\C P^{n-1}$ of lines in $\\C^n$.\n\nNon-example: $G = \\mathrm{SL}_2(\\C)$ and $H$ the diagonal subgroup.\nThen $H$ is closed, but it does not contain any Borel subgroup, since a Borel subgroup of $\\mathrm{SL}_2(\\C)$ has complex dimension $2$ and $H$ has dimension $1$.\nThe quotient $G/H$ is not compact, while every flag manifold is compact.\n\nA flag manifold $G/P$ is compact, simply connected and projective algebraic.\nFor a maximal compact subgroup $K$ of $G$, the group $K$ acts transitively on $G/P$, so $G/P = K/(K \\cap P)$.\nFor $K$ compact connected and $T$ a maximal torus, $K/T$ is the complete flag manifold of the complexification of $K$.\n",
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