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

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 region $\check{F}$ of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over $\check{F}$ (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in $\check{F}$ (discrete orthogonality), arising as the set of points in $\check{F}$ representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus $\mathbb{T}$. The characters of the centre $Z$ of the Lie group allow one to split functions $f$ on $\check{F}$ 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.

arXiv commentPublished in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Weight-Lattice Discretization of Weyl-Orbit Functions
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• On E-Discretization of Tori of Compact Simple Lie Groups: II
• Six types of $E-$functions of the Lie groups O(5) and G(2)
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• Four families of orthogonal polynomials of C2 and symmetric and antisymmetric generalizations of sine and cosine functions
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• Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
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• Spectral measures associated to rank two Lie groups and finite subgroups of $GL(2,\mathbb{Z})$