Separation of variables in the semistable range

Abstract

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 $m\ge 2k-1$ 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.