  In Part 1 of this paper we construct a spectral sequence converging to the
relative Lie algebra cohomology associated to the action of any subgroup $G$ of
the symplectic group on the polynomial Fock model of the Weil representation,
see Section 7. These relative Lie algebra cohomology groups are of interest
because they map to the cohomology of suitable arithmetic quotients of the
symmetric space $G/K$ of $G$. We apply this spectral sequence to the case $G =
\mathrm{SO}_0(n,1)$ in Sections 8, 9, and 10 to compute the relative Lie
algebra cohomology groups $H^{\bullet} \big(\mathfrak{so}(n,1), \mathrm{SO}(n);
\mathcal{P}(V^k) \big)$. Here $V = \mathbb{R}^{n,1}$ is Minkowski space and
$\mathcal{P}(V^k)$ is the subspace of $L^2(V^k)$ consisting of all products of
polynomials with the Gaussian. In Part 2 of this paper we compute the
cohomology groups $H^{\bullet}\big(\mathfrak{so}(n,1), \mathrm{SO}(n); L^2(V^k)
\big)$ using spectral theory and representation theory. In Part 3 of this paper
we compute the maps between the polynomial Fock and $L^2$ cohomology groups
induced by the inclusions $\mathcal{P}(V^k) \subset L^2(V^k)$.
