Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

Abstract

Abstract. The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G. Such functions can always be restricted without loss of information to a fundamental regionF of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated overF (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid inF (discrete orthogonality), arising as the set of points inF representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus T. The characters of the centre Z of the Lie group allow one to split functions f onF into a sum f = f 1 + · · · + f c , where c is the order of Z, and where the component functions f k decompose into the series of C-, or S-, or E-functions from one congruence class only.

E-Orbit Functions

“…As in the case of symmetric and antisymmetric orbit functions, E-orbit functions determine certain orbit function transforms which generalize the Fourier transform (in the case of symmetric orbit functions these transforms generalize the cosine transform and in the case of antisymmetric orbit functions these transforms generalize the sine transform) [7,8,10]. As in the case of symmetric and antisymmetric orbit functions, E-orbit functions determine three types of orbit function transforms: the first one is related to the E-orbit functions E λ (x) with integral λ, the second one is related to E λ (x) with real values of coordinates of λ, and the third one is the related discrete transform.…”

“…If values of λ are integral points lying inside of the fundamental domain F e of the even affine Weyl group W aff e , then the corresponding E-orbit functions are orthogonal on the closure F e of the fundamental domain F e with respect to the Euclidean measure: 10) where the overbar over E λ (x) means complex conjugation. This relation directly follows from the orthogonality of the exponential functions e 2πi µ,x (entering into the definition of E-orbit functions) for different weights µ and from the fact that a given element ν ∈ P belongs to precisely one E-orbit function.…”

“…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.…”

“…The important peculiarity of orbit functions is a possibility of their discretization [10], which is made by using the results of paper [11]. This possibility makes orbit functions useful for applications.…”

“…It is possible to introduce finite E-orbit function transforms, based on E-orbit functions (see [10]). It is done in the same way as in the case of symmetric orbit functions in [7] by using the results of paper [11].…”