I am working on the following problem right now: Let $V,W,H$ be finite dimensional vector spaces. I have a group action $Gl(V)\times Gl(W) \curvearrowright Hom(V\otimes H,W)$ in the obvious way i.e. acting on $V$ first and the acting on $W$. Or more precisly for $G=(A,B) \in Gl(V)\times Gl(W)$ and $C \in Hom(V\otimes H,W)$ we have that
$$ G\cdot C(v\otimes h)=B( C(A^{-1}(v)\otimes h)) $$.
I am now particularly interested in the dimension of the orbits of this action or equivalently the dimension of the stabilizers. This is particularly easy in the case where $H$ is of dimension 1 and I know that the orbits are determined by the rank of the the matrix associated to $C$ in this case.
What I tried concretely is to view the action as action on matrices of the following form
$$(BC_1 A^{-1}\vert \ldots \vert B C_l A^{-1} ) $$
where $l$ is the dimension of $H$ and the $C_i$ are elements of $Mat_{n,m}$. This gives me a lower bound for the dimension by looking at the $C_i$ of maximal rank.
This looks a lot like a "simultaneous" form of row echelon reduction.
My question would be: What is the exact dimension of the orbits? If this is too hard then I'm interested in the dimension of the orbits in the closure of another orbit. In particular does it hold that for $G\cdot y \subset \partial G\cdot x$ that $dim(G\cdot y)\leq dim(G\cdot x)-2$?
I would be very thankfull if somebody who already thought about this problem could provide some insight or if somebody could point me to some literature about this particular problem.