Consider the group action of $G=\mathrm{GL}_2(\mathbb{C})\times \mathrm{SO}_4(\mathbb{C})$ on $M_{2\times 4}(\mathbb{C})$ by $(g,h).x=gxh^{-1}$. What are the orbits of this action? Are still parametrized by rank? Somehow I think the quadratic form should play a role, but I could not figure out how. This might be standard from invariant theory, but I am ignorant of that subject.
Do we know the orbits in the more general case: the action of $\mathrm{GL}_n(\mathbb{C})\times \mathrm{SO}_m(\mathbb{C})$ on $M_{n\times m}(\mathbb{C})$? How about if we replace $\mathrm{SO}_m(\mathbb{C})$ by some other groups, like symplectic group? Are there any standard references on this?
Any comments/reference are welcome. Thanks in advance.
Of course, rank of a matrix is invariant under the $G$-action. So, it is enough to classify orbits on matrices of rank 2, 1, and 0.
In case of rank 2, a $GL_2$-orbit of a matrix is the same as a 2-dimensional subspace in the 4-dimensional space endowed with a non-degenerate quadratic form. Clearly, these orbits are parameterized by the rank of the restriction of the quadratic form, hence there are three of them.
In case of rank 1, a $GL_2$-orbit of a matrix is the same as a 1-dimensional subspace in the 4-dimensional space endowed with a non-degenerate quadratic form. Again, these orbits are parameterized by the rank of the restriction of the quadratic form, hence there are two of them.
Finally, in case of rank 0 there is a single $G$-orbit (just the zero matrix).
So, altogether there are $3 + 2 + 1 = 6$ orbits.