标题： 非对称类型$BC_{1}$的雅可比多项式作为向量值多项式 第1部分：球面函数 显示英文标题

标题： Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions

摘要： 我们通过向量值和矩阵值正交多项式来研究类型$BC_{1}$的非对称雅可比多项式。 作为矩阵值正交多项式的解释，给出了类型$BC_1$的非对称雅可比多项式的新表达式，该表达式以类型$BC_{1}$的对称雅可比多项式为基础。 在这种解释中，以非对称雅可比多项式为本征函数的Cherednik算子对应于类型$BC_{1}$的对称雅可比多项式的两个移位算子。 我们证明，具有所谓几何根重数的类型$BC_{1}$的非对称雅可比多项式，作为向量值多项式解释时，可以与球面$S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$上与$\mathrm{Spin}(2m+1)$的基本自旋表示相关的球函数相识别。 陈德尼克算子在此解释中对应于$S^{2m+1}$上自旋矢的狄拉克算子。 显示英文摘要

摘要： We study non-symmetric Jacobi polynomials of type $BC_{1}$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type $BC_1$ in terms of the symmetric Jacobi polynomials of type $BC_{1}$. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type $BC_{1}$. We show that the non-symmetric Jacobi polynomials of type $BC_{1}$ with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere $S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$ associated with the fundamental spin-representation of $\mathrm{Spin}(2m+1)$. The Cherednik operator corresponds to the Dirac operator for the spinors on $S^{2m+1}$ in this interpretation.