New expressions for the eigenvalues of the invariant operators of O(N) and Sp(2n)

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Abstract
In the spirit of the recent work of Popov for u(n), we derive a direct expansion of the eigenvalues of the invariant operators Cp for the orthogonal and symplectic groups in terms of the power sums with completely specified coefficients βp(ν), which are easy to compute. The resulting expression, which is a complete analog of the u(n) results, is closed, simple and manifests the general structure of the Cp. It is now possible to say for what value of p a particular combination of the Sk’s begin to appear. Explicit applications of these results in computing the Cp for p<8 illustrate fully their simplicity. Thus our work simplifies and unifies the treatment of this aspect of the problem for the semisimple Lie groups.