Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 006, 60 pages      math-ph/0601037      https://doi.org/10.3842/SIGMA.2006.006

Orbit Functions

Anatoliy Klimyk a and Jiri Patera b
a) Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., Kyiv, 03143 Ukraine
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received January 04, 2006; Published online January 19, 2006

Abstract
In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space En 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 En. Orbit functions are solutions of the corresponding Laplace equation in En, 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.

Key words: orbit functions; Coxeter-Dynkin diagram; Weyl group; orbits; products of orbits; orbit function transform; finite orbit function transform; Neumann boundary problem; symmetric polynomials.

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