Geodesic
Orbit Functions
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: 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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Anatoliy Klimyk; Jiri Patera. Orbit Functions. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a5/

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