Casimir operators for semisimple Lie groups

A simple method is developed for computing the eigenvalues of the invariant operators (the so-called Casimir operators) $\widehat C_p$ of arbitrary order $p$ for semisimple Lie groups. The resulting formulas (52) and (55) are applicable for the case that among the irreducible representations of the given group there is at least one representation with a simple spectrum – in particular, for all the classical groups, as well as the groups $G_2$, $E_6$ and $E_7$. An expression is found (see (75)) for the generating function of the Casimir operators in the case of the classical groups.