Orbit Functions of Compact Semisimple Lie Groups as Special Functions

An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. Properties of such functions will be described for compact simple Lie groups of all types. In particular, products of the functions decompose into their sums, the functions are periodic on copies of the fundamental region F of G, they are solutions of corresponding Laplace equation in n dimensions, satisfying Neumann condition at the boundary of F , . . . Uncommon applications to image enhancements and a new approach to n-dimensional data compression are shown, when orbit functions are evaluated at suitable sets of conjugacy classes of elements of finite order in G.