Separation of variables in the semistable range

2019/02/07 by R. Lavicka, Lavicka, Roman
Physics and Astronomy · #17B10 #30G35 #Black Holes and Theoretical Physics #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1902.02555

openalex publication_date 2019/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we give an alternative proof of separation of variables for scalar-valued polynomials P:(\mathbb R^m)^k→\mathbb C in the semistable range m≥ 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.

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