  Let $\Gamma_X$ denote the space of all locally finite configurations in a
complete, stochastically complete, connected, oriented Riemannian manifold $X$,
whose volume measure $m$ is infinite. In this paper, we construct and study
spaces $L^2_\mu\Omega^n$ of differential $n$-forms over $\Gamma_X$ that are
square integrable with respect to a probability measure $\mu$ on $\Gamma_X$.
The measure $\mu$ is supposed to satisfy the condition $\Sigma_m'$ (generalized
Mecke identity) well known in the theory of point processes. On
$L^2_\mu\Omega^n$, we introduce bilinear forms of Bochner and deRham type. We
prove their closabilty and call the generators of the corresponding closures
the Bochner and deRham Laplacian, respectively. We prove that both operators
contain in their domain the set of all smooth local forms. We show that, under
a rather general assumption on the measure $\mu$, the space of all
Bochner-harmonic $\mu$-square integrable forms on $\Gamma_X$ consists only of
the zero form. Finally, a Weitzenb\"ock type formula connecting the Bochner and
deRham Laplacians is obtained. As examples, we consider (mixed) Poisson
measures, Ruelle type measures on $\Gamma_{{\Bbb R}^d}$, and Gibbs measures in
the low activity--high temperature regime, as well as Gibbs measures with a
positive interaction potential on $\Gamma_X$.
