# Separation of variables in the semistable range

## TL;DR

This paper presents an alternative proof for the separation of variables in scalar-valued polynomials within the semistable range, linking polynomial decomposition to representation theory of orthogonal and symplectic groups.

## Contribution

It introduces a novel proof method connecting polynomial decomposition to irreducibility of generalized Verma modules, with potential applications to spinor-valued polynomials.

## Key findings

- Proof of separation of variables in the semistable range
- Equivalence of polynomial decomposition uniqueness and module irreducibility
- Potential extension to spinor-valued polynomials

## 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 $m\geq 2k-1$ 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.