---
tag: JC9R1073M9
kind: definition
name: Flag manifold
msc: [14M15, "22E46", 53C30]
defines: BB0TSMW93P
uses: [7JBRGCVJRB, XJS07AJ50Y, 5JSYA1JNCC, 85TWR6E75P]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_flag_variety
---

## Definition

Let $G$ be a connected complex [[semisimple Lie group|XJS07AJ50Y]].
Let $B$ be a [[Borel subgroup|5JSYA1JNCC]] of $G$, a maximal connected solvable closed complex subgroup.
Let $P$ be a closed complex subgroup of $G$ with $B \subseteq P$, a parabolic subgroup.
The *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.
When $P = B$, the space $G/B$ is the *complete* (or full) flag manifold of $G$.

## Remarks

Example: $G = \mathrm{SL}_n(\C)$ with $B$ the upper triangular matrices of determinant $1$.
Then $G$ 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 $V_i = \operatorname{span}(e_1, \dots, e_i)$.
So $G/B$ is the space of complete flags, of complex dimension $n(n-1)/2$.
The 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 $\C^n$.

Non-example: $G = \mathrm{SL}_2(\C)$ and $H$ the diagonal subgroup.
Then $H$ 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 $1$.
The quotient $G/H$ is not compact, while every flag manifold is compact.

A flag manifold $G/P$ is compact, simply connected and projective algebraic.
For a maximal compact subgroup $K$ of $G$, the group $K$ acts transitively on $G/P$, so $G/P = K/(K \cap P)$.
For $K$ compact connected and $T$ a maximal torus, $K/T$ is the complete flag manifold of the complexification of $K$.