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

SIGMA 3 (2007), 023, 83 pages      math-ph/0702040      https://doi.org/10.3842/SIGMA.2007.023

Antisymmetric 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 December 25, 2006; Published online February 12, 2007

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

Key words: antisymmetric orbit functions; signed orbits; products of orbits; orbit function transform; finite orbit function transform; finite Fourier transforms; finite cosine transforms; finite sine transforms; symmetric functions.

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