Tangent space of a flag manifold? I am studying differential geometry and now I am trying to find the tangent space to a flag manifold $F(a_1,a_2,...,a_k, \mathbb{R^n}).$
Any hint would be greatly appreciated!
 A: Try to realise your flag manifold as a homogeneous space $G / P$ for some Lie group $G$. Then the tangent space is $\mathfrak{g} /\mathfrak{p}$ where $\mathfrak{g}$ and $\mathfrak{p}$ are the Lie algebras of $G$ and $P$, respectively.
A: Let $\mathbb{V}$ be a finite dimensional real (Euclidean) or complex (Hermitian) vector space. Let $G(r ; \mathbb{V})$ denote the Grassmann manifold of $r$ dimensional linear subspaces of $\mathbb{V}$, which is
$$
G(r ; \mathbb{V}):=\{\text { all subspaces } E \subset \mathbb{V} \mid \operatorname{dim}(E)=r\}.
$$
Let $a_{\bullet}$ be an increasing of positive integers
$$
a_{\bullet}=\left(1 \leq a_1<a_2<\cdots<a_k<\operatorname{dim}(\mathbb{V})\right).
$$
Define the flag manifold $F\left(a_{\bullet} ; \mathbb{V}\right)$ by
$$
F\left(a_{\bullet} ; \mathbb{V}\right):=\left\{\left(E_1, \ldots, E_k\right) \in G\left(a_1 ; \mathbb{V}\right) \times G\left(a_2 ; \mathbb{V}\right) \times \cdots \times G\left(a_k ; \mathbb{V}\right) \mid E_i \subset E_{i+1}\right\}.
$$
Let $\left(E_1, \ldots, E_k\right)$ be a point of $F\left(a_{\bullet} ; \mathbb{V}\right)$, and the tangent space to $F\left(a_{\bullet} ; \mathbb{V}\right)$ at this point will be as subspace of
$$
\operatorname{Hom}\left(E_1, E_1^{\perp}\right) \oplus \operatorname{Hom}\left(E_2, E_2^{\perp}\right) \oplus \cdots \oplus \operatorname{Hom}\left(E_k, E_k^{\perp}\right).
$$
To show this we only need basic knowledge of linear algebra.
