Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one has $v \in V$, and $H \subset G$ is the stabilizer of $V$, then $H$ is a Lie subgroup of $G$ and there is an injective immersion $G/H \rightarrow V$ whose image is the orbit of $v$. (This is true generally for a smooth action). Is this map always an embedding?
I know that the orbits need not be closed submanifolds of $V$.
If this is not true, are there straightforward conditions under which one can guarantee that it is true? Certainly $G$ compact is enough, but this is too strong. $G$ reductive, perhaps?