Antisymmetric Orbit Functions

Abstract

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

E-Orbit Functions

“…Using the explicit formula for the antisymmetric orbit function ϕ ρ (x), where ρ is a half of positive roots, we have derived in [8] explicit forms of the fundamental domains of W aff for the cases A n , B n , C n , D n . They easily determine fundamental domains F e for the corresponding even affine Weyl groups W aff e .…”

“…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].…”

“…This possibility makes orbit functions useful for applications. In particular, on this way multivariate discrete Fourier transform and multivariate discrete sine and cosine transforms are received (see [8,12,13]). …”