$GL_{n}(\mathbb C) \times GL_{m}(\mathbb C)$ acts on the space $P$ of polynomial functions on the space of complex $n$ by $m$ matrices like this:
$((g,h)f)(A)=f(g^{T}Ah) $ for $g \in GL_{n}(\mathbb C)$, $h \in GL_{m}(\mathbb C)$, $f \in P$ and $A$ a complex $n$ by $m$ matrix.
I would like to show that the degree $d$ homogeneous polynomials form an invariant subspace, and that the induced representation of $GL_{n}(\mathbb C) \times GL_{m}(\mathbb C)$ on $P_{d}$ is a polynomial representation.
The first part seems too easy: if $f$ is homogenous of degree $d$, then the image under the action will again be such a polynomial. Am I missing something here?
For the representation part I don't have an idea besides that I need to show that the elements of the representation are restrictions to $GL_{n}(\mathbb C) \times GL_{m}(\mathbb C)$ of polynomials on $\mathbb C^{nxn} \oplus \mathbb C^{mxm}$