  In this paper, we give an alternative proof of separation of variables for
scalar-valued polynomials $P:(\mathbb R^m)^k\to\mathbb C$ in the semistable
range $m\geq 2k-1$ for the symmetry given by the orthogonal group $O(m)$. It
turns out that uniqueness of the decomposition of polynomials into spherical
harmonics is equivalent to irreducibility of generalized Verma modules for the
Howe dual partner $sp(2k)$ generated by spherical harmonics. We believe that
this approach might be applied to the case of spinor-valued polynomials and to
other settings as well.
