Laplacian and homology of free two-step nilpotent Lie algebras

In this paper we determine the homology (with trivial coefficients) of the free two-step nilpotent Lie algebras over the complex numbers. This is done by working out the structure of the homology as a module under the general linear group. The main tool is a Laplacian for the free two-step nilpotent Lie algebras, which turns out to be closely related to the Casimir operator of the general linear group. We are able to compute all eigenvalues of the Laplacian on the chain complex. (C) 1996 Academic Press, Inc.