Flag manifold

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 $K$.