  It is shown, for any irreducible representation of $E_8$ Lie algebra, that
eigenvalues of Casimir operators can be calculated in the form of invariant
polinomials which are decomposed in terms of $A_8$ basis functions. The general
method is applied for degrees 8,12 and 14 for which 2,8 and 19 invariant
polinomials are obtained respectively. For each particular degree, these
invariant polinomials can be taken to be $E_8$ basis functions in the sense
that any Casimir operator of $E_8$ has always eigenvalues which can be
expressed as linear superpositions of them. This can be investigated by showing
that each one of these $E_8$ basis functions gives us a linear equation to
calculate weight multiplicities.
