$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$
Let $V$ be a real $d$-dimensional vector space ($d \ge 4$). Let $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \text{End}(V) \mid \operatorname{rank}(A) > k \}$. $H_{>k}$ is an open connected submanifold of $ \text{End}(V)$.
Question: Does $H_{>k}$ admit a transitive Lie group action?
That is, does there exist a (finite dimeniosnal) Lie group $G$, and a smooth action of $G$ on $H_{>k}$, such that $H_{>k}$ itself is the single orbit?
There is a natural action on $H_{>k}$ by $\GL(V) \times \GL(V)$, given by $$ (A,B) \cdot C=ACB^{-1}. $$
This action has a finite number of orbits; each orbit is the space of endomorphisms of a certain rank. So, it is not transitive. (but is "close" to being one).
Even if there is a transitive action, I don't expect it to be as "natural" as the action considered above. Still, I am interested to know whether or not such group action exist.
I know there are obstructions for a manifold to admit a transitive Lie group action in the case where it's compact. Here, however, our manifold $H_{>k}$ is non-compact.
