厄米矩阵单纯形上的雅可比多项式与部分旗流形上的带状球函数
Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
AI总结：
研究部分旗流形上拉普拉斯 - 贝尔特拉米算子径向部分的本征函数，利用矩阵变量孔恩温德方法，由厄米雅可比多项式构造正交多项式并猜想其为基本带状球函数，还借助多元舒尔多项式获取相关信息及构造函数。
AI中文摘要：
我们引入了部分旗流形上拉普拉斯 - 贝尔特拉米算子的径向部分，并研究其本征函数，即基本带状球函数。这些本征函数可与酉矩阵同时共轭下厄米矩阵单纯形上关于复矩阵变量狄利克雷分布不变的正交多项式相关。利用矩阵变量版本的孔恩温德方法，从单变量正交多项式构造多变量正交多项式，我们用厄米雅可比多项式构造了一族两两正交的多项式，并猜想它们是基本带状球函数。此外，我们回顾了多元舒尔多项式的定义，并展示了如何用它们获取基本带状球函数的信息。特别是，我们给出了关于部分旗流形上拉普拉斯 - 贝尔特拉米算子的径向部分相对于多元舒尔多项式是强三角的一些部分结果。然后通过对多元舒尔函数应用格拉姆 - 施密特正交化过程来构造基本带状球函数。
英文摘要：
We introduce the radial part of the Laplace--Beltrami operator on partial flag manifolds and study its eigenfunctions, the elementary zonal spherical functions. These eigenfunctions can be related to orthogonal polynomials on the simplex of Hermitian matrix invariant under simultaneous conjugation by unitary matrices with respect to the complex matrix variate Dirichlet distribution. Using a matrix variate version of Koornwinders method for constructing orthogonal polynomials in multiple variables from orthogonal polynomials in one variable, we construct a family of pairwise orthogonal polynomials in terms of Hermitian Jacobi polynomials and conjecture that these are elementary zonal spherical functions. Furthermore, we recall the definition of multivariate Schur polynomials and show how they can be used to obtain information about elementary zonal spherical functions. In particular, we give some partial results in the direction that the radial part of the Laplace--Beltrami operator on partial flag manifolds is strongly triangular with respect to the multivariate Schur polynomials. The elementary zonal spherical functions can then be constructed by applying the Gram--Schmidt orthogonalisation process to the multivariate Schur functions.
