Mathematical Sciences: Symmetry and Orthogonal Polynomials

我们将研究特殊函数理论中的几个领域。特别的重点将放在几个变量的正交多项式的数学理论的发展，其中将有一个集中的函数定义在n球。取这些多项式与有限反射群下不变的权函数正交。权函数是线性函数乘积的幂，这些线性函数的零集是群中反射的镜像。Selberg积分的一种形式，对应于八面体群，是一个重要的特例。利用一阶微分-差分算子的交换代数，其作用类似于球谐波理论中的偏导数，工作将集中在三个问题上。首先是构造一个常微分算子和微分-差分算子交织在一起的非贝尔分数积分算子。这给出了拉普拉斯函数的调和函数和其他特征函数到所研究的函数的映射。其次，利用这些算子及其伴随，构造球面上函数的显式正交基和范数。重点将首先放在三维和四维的例子。第三，我们将把最近关于球的泊松核的正性结果推广到在一般反射群中明确地找到这个核。这项工作有望应用于理论物理学。它已经证明了它的实用性，在新的凯克10米望远镜的镜面研磨中发挥了重要作用。

Several areas within the theory of special functions will be investigated. Particular emphasis will be placed on the development of a mathematical theory of orthogonal polynomials of several variables in which there will be a concentration on functions defined on the n-sphere. These polynomials are taken to be orthogonal with respect to a weight function which is invariant under a finite reflection group. The weight functions are powers of products of linear functions whose zero sets are the mirrors for the reflections in the group. A form of Selberg's integral, corresponding to the octahedral group, is an important special case. By means of a commutative algebra of first-order differential-difference operators, which have an action analogous to that of partial derivatives in the theory of spherical harmonics, work will focus on three problems. The first is to construct a nonabelian fractional integral operator which intertwines ordinary differentiation and the differential- difference operators. This gives a mapping from harmonic functions and other eigenfunctions of the Laplacian to the functions under investigation. Secondly, by use of these operators and their adjoints, work will be done constructing explicit orthogonal bases and norms for functions on the sphere. Emphasis will be placed first on three and four dimensional examples. Thirdly, work will be done extending a recent positivity result on the Poisson kernel for the ball to finding this kernel explicitly for the general reflection group. This work is expected to have application to theoretical physics. It has already demonstrated its utility by playing a major role in the grinding of mirrors for the new Keck 10-meter telescope.