轨道函数
Orbit Functions
摘要：
本文对轨道函数的性质进行了回顾和发展。欧几里德空间$E_n$上的轨道函数是对称化的指数函数。对称化由对应于Coxeter-Dynkin图的Weyl群实现。将描述这样的函数的属性。轨道函数是秩为$n$的紧致半单李群$G$从其Weyl群轨道之一对不可约特征标的贡献。证明了在整个欧氏空间$E_n$中，轨道函数的值在仿射Weyl群（由初始Weyl群确定）的基本域$F$的副本上重复.轨道函数是$E_n$中相应拉普拉斯方程的解，满足$F$边界上的Neumann条件。轨道函数确定对称傅里叶变换和有限点集上的变换。
英文摘要：
In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space $E_n$ are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group $G$ of rank $n$ from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Orbit functions are solutions of the corresponding Laplace equation in $E_n$, satisfying the Neumann condition on the boundary of $F$. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.
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。
