I am working on the space $V_k (\mathbb{C}^n) = \left\lbrace (v_1, \cdots , v_k ) \in (\mathbb{C}^n)^k | \langle v_i, v_j \rangle = \delta_{ij} \right\rbrace $.
I define the continuous action of $U(n)$ on $V_k (\mathbb{C}^n)$ by $ U \cdot (v_1, \cdots , v_k ) = (U v_1, \cdots , U v_k ) $ .
I know that it's transitive since I can find a matrix $ U \in U(n)$ that takes me from a vector with norm 1 to another vector with norm 1 ( Is this enough? Can this be made more rigorous?)
I want to show that the stabilizer of any element of $V_k (\mathbb{C}^n)$ is isomorphic to $U(n-k)$; but I can't really see it!
Here is a hint:
Suppose $v \in (\mathbb{C}^n)^k$, and let $v_{k+1},...,v_n$ be such that $v_1,...,v_n$ is an orthonormal basis of $\mathbb{C}^n$. Let $V = \operatorname{sp} \{v_1,...,v_k \}$.
Suppose $U \in G_v$, and note that $U V = V$ and $U V^\bot = V^\bot$, which means that in the basis $v_1,...v_n$, $U$ has the form $U = \begin{bmatrix} I & 0 \\ 0 & \tilde{U} \end{bmatrix}$, where $\tilde{U}: V^\bot \to V^\bot$.