E-Orbit Functions

TL;DR

This paper reviews and advances the theory of E-orbit functions, which are symmetrized exponential functions related to Weyl groups, and explores their properties and applications in Fourier transforms.

Contribution

It introduces new properties of E-orbit functions and develops their role as kernels in symmetrized Fourier transforms and discrete transforms on fundamental domains.

Findings

E-orbit functions are invariant under the even part of affine Weyl groups.

They serve as kernels for symmetrized Fourier transforms.

The paper establishes a discrete E-orbit function transform on finite point sets.

Abstract

We review and further develop the theory of $E$-orbit functions. They are functions on the Euclidean space $E_n$ obtained from the multivariate exponential function by symmetrization by means of an even part $W_{e}$ of a Weyl group $W$, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group $W$. The $E$-orbit functions, determined by integral parameters, are invariant with respect to even part $W^{\rm aff}_{e}$ of the affine Weyl group corresponding to $W$. The $E$-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain $F^{e}$ of the group $W^{\rm aff}_{e}$ (the discrete $E$-orbit function transform).