Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).

There comes a point in the paper (Lemma 2.8) when he uses the fact that the ring of all constant coefficient $SO(n)$-invariant differential operators on $M_{n,m}$ (viewed as $\mathbb R^{nm}$) without constant term is generated by the operators $$\Delta_{il}=\sum_{k=1}^{n} \frac{\partial}{\partial x_{ki}}\frac{\partial}{\partial x_{kl}} \;\;\;\;\;\;\;1 \leq i, l \leq m$$ where $m < n$.

I don't see how the ring is generated by operators of this form. How can we show this?

• Aren't you assuming that the coefficients are constant? After all, the first order operator that is differentiation in the radial direction is invariant under $\mathrm{GL}(mn,\mathbb{R})$ and it's not generated by the $\Delta_{il}$. Once you do assume that it has constant coefficients, you can reduce it to the question of what are the $\mathrm{SO}(n)$-invariant polynomials, and that's not hard. – Robert Bryant Jul 13 '15 at 18:59
• I am--thanks for pointing that out. – Ken Schefers Jul 13 '15 at 19:29