New families of Weyl group orbit functions

Details New families of Weyl group orbit functions New families of Weyl group orbit functions

2012-02-20

arxiv.org/abs/1202.4415v1

Physics

Mathematical Physics

13 pages

Scientific paper

Two families of Weyl group-invariant functions, denoted C, S, are known to arise from the irreducible representations of the compact simple Lie groups, and they have been studied extensively. These special functions derive respectively from Weyl orbits and from alternating Weyl orbits associated with the sign character of the Weyl group. In the case that the roots of the Lie group are of two different lengths, there are mixed sign characters and they give rise to two new families of functions, the S^L and S^S families. The subject of the paper is the definition and properties of these new functions. We describe their symmetry properties, their continuous orthogonality over the fundamental region of the affine Weyl group of the corresponding simple Lie algebra, and their behaviour at the boundary of the fundamental region. This involves a study of the subroot systems composed respectively of the long and short roots, and on the way in which their Weyl groups fit into the Weyl group of the original roots system.

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