  It is given a way of computing Casimir eigenvalues for Weyl orbits as well as
for irreducible representations of Lie algebras. A kappa(s) number of
polinomials which depend on rank N are obtained explicitly for A_N Casimir
operators of order s where kappa(s) is the number of partitions of s into
positive integers except 1. It is also emphasized that these eigenvalue
polinomials prove useful in obtaining formulas to calculate weight
multiplicities and in explicit calculations of the whole cohomology ring of
Classical and also Exceptional Lie algebras.
