I'm trying to prove that the little group of $O(n)$ acting on a $k$-dimensional subspace of $\mathbb{R}^n$, call it $V$, is $O(k)\times O(n - k)$ due to the Grassmann manifold is isomorphic to $O(n)/(O(k)\times O(n - k))$.
I tried following the next steps: for elements in the little group, call them $g_l$, $q \in V$ has to be invariant, that is $g_lq = q$ and therefore:
$$g_l = \begin{pmatrix}1_k & 0 \\0 & A_{n - k} \end{pmatrix} \tag1$$
Where $1_k$ is the $k\times k$ identity matrix and $A_{n-k}$ a $(n - k)\times (n - k)$ matrix. A similar process can be seen in Isotropy group of $SO(n)$. I presume that $A_{n - k} \in O(n - k)$ because we are working with orthonormal transformations.
Therefore, we conclude that the little group is $1_k\times O(n - k)$ which is different from the deduction by the isomorphism for Grassmann manifold.
I'm doing something bad but I don't know what. Can you show me the way?