Two families of Weyl group-invariant functions, denoted C, S, are known toView original: http://arxiv.org/abs/1202.4415
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.