半稳定区间的分离变量
Separation of variables in the semistable range
摘要：
本文对正交群$O(m)$给出对称性的标量值多项式$P:(\mathbb R^m)^k\to\mathbb C$在半稳定域$m\geq 2k-1$中的分离变量给出了另一种证明.证明了球面调和多项式分解的唯一性等价于球面调和生成的Howe对偶$sp(2k)$的广义韦尔马模的不可约性.我们相信，这种方法可能适用于旋值多项式的情况下，以及其他设置。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
