Abstrakt
In this paper, we give an alternative proof of separation of variables for scalar-valued polynomials in the semistable range for the symmetry given by the orthogonal group. It turns out that uniqueness of the decomposition of polynomials into spherical harmonics is equivalent to irreducibility of generalized Verma modules for the symplectic algebra 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.