反对称轨道函数
Antisymmetric Orbit Functions
摘要：
本文对反对称轨道函数的性质进行了回顾和发展。欧氏空间$E_n$上的反对称轨道函数是反对称化的指数函数。反对称化由Weyl群实现，对应于Coxeter-Dynkin图。描述了这些函数的性质。这些函数与秩为$n$的紧致半单李群$G$的不可约特征标密切相关。直到一个符号，反对称轨道函数的值在整个欧氏空间$E_n$中仿射Weyl群（由初始Weyl群确定）的基本域$F$的副本上重复。反对称轨道函数是$E_n$中相应拉普拉斯方程的解，在基本域$F$的边界上消失。反对称轨道函数确定了一个所谓的反对称傅里叶变换，它与中心函数在群$G$的不可约表示特征标中的展开密切相关。它们还确定了$F$（离散反对称轨道函数变换）的有限点集上的变换。给出了对称和反对称多元指数、正弦和余弦离散变换。
英文摘要：
In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric 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 $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.
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