  After defining cohomologically higher order BRST and anti-BRST operators for
a compact simple algebra {\cal G}, the associated higher order Laplacians are
introduced and the corresponding supersymmetry algebra $\Sigma$ is analysed.
These operators act on the states generated by a set of fermionic ghost fields
transforming under the adjoint representation. In contrast with the standard
case, for which the Laplacian is given by the quadratic Casimir, the higher
order Laplacians $W$ are not in general given completely in terms of the
Casimir-Racah operators, and may involve the ghost number operator. The higher
order version of the Hodge decomposition is exhibited. The example of su(3) is
worked out in detail, including the expression of its higher order Laplacian W.
