On Discretization of Tori of Compact Simple Lie Groups II

The discrete orthogonality of special function families, called $C$- and $S$-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivn'ak and Patera (2009 J. Phys. A: Math. Theor. 42 385208). Here, the results of Hrivn'ak and Patera are extended to two additional recently discovered families of special functions, called $S^s-$ and $S^l-$functions. The main result is an explicit description of their pairwise discrete orthogonality within each family, when the functions are sampled on finite fragments $F_M^s$ and $F_M^l$ of a lattice in any dimension $n\ge 2$ and of any density controlled by $M$, and of the symmetry of the weight lattice of any compact simple Lie group with two different lengths of roots.