Orbit Functions

TL;DR

This paper reviews and advances the theory of orbit functions, which are symmetrized exponential functions related to Weyl groups, and explores their properties, solutions to Laplace's equation, and applications in Fourier transforms.

Contribution

It provides a comprehensive review and new insights into the properties and applications of orbit functions in Lie group theory and harmonic analysis.

Findings

Orbit functions are solutions to Laplace's equation with Neumann boundary conditions.

Values of orbit functions repeat across fundamental domains of affine Weyl groups.

Orbit functions enable symmetrized Fourier transforms and finite point transforms.

Abstract

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.