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This is a question from Lee's book on Smooth Manifolds, question 21-16:

Let $F_K(V)$ be the set of flags of type $K$ in a finite-dimensional (real) vector space $V$. Show that $GL(V)$ acts transitively on $F_K(V)$, and that the isotropy group of a particular flag is a closed subgroup of $GL(V)$, and conclude that $F_K(V)$ has a unique smooth manifold structure such that the action is smooth. What is the dimension of $F_K(V)$? For which $K$ is $F_K(V)$ compact?

I am not sure how to do compactness.

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I read from Lee - Introduction to Smooth Manifolds, example 7.23, by flag of type $K$ of a finite dimensional real vector space $\mathbb{V}$ of dimension $n$ he means \begin{gather} 0<k_1<k_2<\dots<k_m\leq n\in\mathbb{N},\,K=(k_1,\dots,k_m),\\ F\equiv\{\underline0\}<\mathbb{V}_1<\mathbb{V}_2<\dots<\mathbb{V}_m\leq\mathbb{V},\,\forall i\in\{1,\dots,m\},\,\dim\mathbb{V}_i=k_i,\,dim\mathbb{V}_{i\displaystyle/\mathbb{V}_{i-1􀀀}}=d_i, \end{gather} and $F_K(\mathbb{V})$ is the set of all flags of $\mathbb{V}$ of type $K$.

Let $\{e_1\dots,e_n\}$ be a basis of $\mathbb{V}$ such that: $$ \forall i\in\{1,\dots,m\},\,\mathbb{V}_i=\langle e_1,\dots,e_{k_i}\rangle; $$ then $\forall M\in\mathrm{GL}(n,\,\mathbb{R}),i\in\{1,\dots,m\},\,M\cdot\mathbb{V}_i=\langle M\cdot e_1,\dots,M\cdot e_{k_i}\rangle$ are well defined subspaces of $\mathbb{V}$; in particular, the flag $$ F_K(\mathbb{V})\ni M\cdot F\equiv\mathbb{V}_0< M\cdot\mathbb{V}_1<\dots<M\cdot\mathbb{V}_m\leq\mathbb{V} $$ is well defined.

After all this, one can define the action $$ \alpha:(M,F)\in\mathrm{GL}(n,\mathbb{R})\times F_K(\mathbb{V})\to M\cdot F\in F_K(\mathbb{V}); $$ one (easily) prove that $\alpha$ is a transitive action and therefore $F_K(\mathbb{V})$ is in bijection with $$ \mathrm{GL}(n,\mathbb{R})_{\displaystyle/P(d_1,\dots,d_m)}, $$ where $P(d_1,\dots,d_m)$ is the (closed) subgroup of $\mathrm{GL}(n,\mathbb{R})$ generated by the matrices of the following type $$ \forall i\leq j\in\{1,\dots,m\},\,A_i^j\in\mathbb{R}_{d_i}^{d_j},\,\begin{pmatrix} A_1^1 & A_1^2 & \dots & A_1^r\\ \underline{0}_{d_2}^{d_1} & A_2^2 & \dots & A_2^m\\ \vdots & \ddots & \ddots & \vdots\\ \underline{0}_{d_m}^{d^1} & \underline{0}_{d_m}^{d_2} & \dots & A_m^m \end{pmatrix}. $$ In particular, one can restrict the focus on the isometries of $\mathbb{V}$; one can define the action $$ \widetilde{\alpha}:(M,F)\in\mathrm{O}(n,\mathbb{R})\times F_K(\mathbb{V})\to M\cdot F\in F_K(\mathbb{V}) $$ and as for $\alpha$, one prove that $\widetilde{\alpha}$ is a transitive action and therefore $F_K(\mathbb{V})$ is in bijection with $ \mathrm{O}(n,\mathbb{R})_{\displaystyle/\mathrm{O}(d_1,\mathbb{R})\times\dots\mathrm{O}(d_m,\mathbb{R})}$; because $\mathrm{O}(n,\mathbb{R})$ is a compact manifold and a quotient space of a compact space is compact: $F_K(\mathbb{V})$ is a compact manifold.

Is it all clear?

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