Orbit Functions

Abstract

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.

E-Orbit Functions

“…The group W aff is a semidirect product of its subgroups W andQ ∨ , whereQ ∨ is an invariant subgroup (see Section 5.2 in [7] for details).…”

“…We have given this information in [7] and [8]. In order to make this paper selfcontained we repeat shortly a part of that information in Section 2.…”

“…We shall call these functions E-orbit functions, since they are an analogue of the well-known exponential function (since symmetric and antisymmetric orbit functions φ λ and ϕ λ are generalizations of the cosine and sine functions, they are often called as C-functions and S-functions, respectively). This paper is a natural continuation of our papers [7] and [8]. Under our exposition we use the results of papers [2] and [10], where E-orbit functions are defined.…”

“…Orbit functions on the 2-dimensional Euclidean space E 2 , invariant or anti-invariant with respect to W , were considered in detail in [3,4,5,6]. A detailed description of symmetric and antisymmetric orbit functions on any Euclidean space E n is given in [7] and in [8]. Orbit functions on E 2 , invariant with respect to W e are studied in [9].…”

“…Being a modification of monomial symmetric functions, they are directly related to the theory of symmetric (Laurent) polynomials [14,15,16,17] (see Section 11 in [7]). …”