Centre de recherches mathematiques,

Universite de Montreal,

C.P. 6128 succursale Centre-ville,

Montreal (Quebec) H3C 3J7, CANADA

E-mail: patera@crm.umontreal.ca

**Orbit functions of compact semisimple Lie groups as
special functions**

**Abstract:**

An orbit function is the contribution to the character of an irreducible
finite dimensional representation 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 eigenfunctions of corresponding Laplace operators 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 will be shown, when orbit functions are evaluated at suitable
sets of conjugugacy classes of elements of finite order in *G*.