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$GL_{n}(\mathbb C) \times GL_{m}(\mathbb C)$ 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 $\mathbb C^{nxn} \oplus \mathbb C^{mxm}$

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You are not missing anything on the first part. – darij grinberg Jan 13 '12 at 21:36
About the second part: You have to prove that, for every $f\in P$ and any linear form $\zeta: P\to\mathbb C$, the value of $\zeta\left(A\mapsto f\left(g^TAh\right)\right)$ is a polynomial function in the entries of the matrices $g$ and $h$. This isn't difficult. (Notice how linear forms $\zeta: P\to\mathbb C$ look like: each of them maps every polynomial to some fixed $\mathbb C$-linear combination of its coefficients.) – darij grinberg Jan 13 '12 at 21:39

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