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2部分：移位算子 显示英文标题

标题： Non-symmetric Jacobi polynomials of type $BC_1$ as vector-valued polynomials Part 2: Shift operators

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使我们能够为非对称雅可比多项式引入位移算子。 这些位移算子是微分-反射算子，我们给出了四个基本的位移算子，它们在生成所有位移算子的意义上是基本的。 此外，这些基本位移算子的对称化是类型$BC_1$的对称雅可比多项式的基本位移算子。 为了将类型$BC_1$的非对称雅可比多项式实现为不变的$\mathbb{C}^2$值的洛朗多项式，我们引入了一个同态，它类似于类型$BC_1$的对称雅可比多项式的哈里什-钱德拉同态。 对于几何根重数，类型$BC_1$的非对称雅可比多项式可以解释为球函数，我们证明了在此上下文中我们的哈里什-钱德拉同态通过径向部分映射与莱波夫斯基同态相关。 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 allows us to introduce shift operators for the non-symmetric Jacobi polynomials. The shift operators are differential-reflection operators and we present four of these operators that are fundamental in the sense that they generate all shift operators. Moreover, the symmetrizations of these fundamental shift operators are the fundamental shift operators for the symmetric Jacobi polynomials of type $BC_1$. For the realization of non-symmetric Jacobi polynomials of type $BC_1$ as invariant $\mathbb{C}^2$-valued Laurent polynomials, we introduce a homomorphism that is analogous to the Harish-Chandra homomorphism for the symmetric Jacobi polynomials of type $BC_1$. For geometric root multiplicities, the non-symmetric Jacobi polynomials of type $BC_1$ can be interpreted as spherical functions and we show that our Harish-Chandra homomorphism in this context is related to the Lepowsky homomorphism via the radial part map.