Orbit Functions

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.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (2)

• Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
• Compact Simple Lie Groups and Their C-, S-, and E-Transforms

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• Orthogonal polynomials of compact simple Lie groups
• Four families of Weyl group orbit functions of B_3 and C_3
• Polytope Expansion of Lie Characters and Applications
• Spectral measures associated to rank two Lie groups and finite subgroups of $GL(2,\mathbb{Z})$
• On immanant functions related to Weyl groups of $A_n$