Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 076, 14 pages      math-ph/0611020      https://doi.org/10.3842/SIGMA.2006.076

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

Robert V. Moody a and Jiri Patera b
a) Department of Mathematics, University of Victoria, Victoria, British Columbia, Canada
b) Centre de Recherches Mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal, H3C 3J7, Québec, Canada

Received October 30, 2006; Published online November 08, 2006

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 F of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over F (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in F (discrete orthogonality), arising as the set of points in F 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 on F into a sum f = f1 + ¼ + fc, where c is the order of Z, and where the component functions fk decompose into the series of C-, or S-, or E-functions from one congruence class only.

Key words: orbit functions; Weyl group; semisimple Lie group; continuous orthogonality; discrete orthogonality.

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References

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