The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)

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 $G\mathrm{/}K$ of $G$. We apply this spectral sequence to the case $G = $\mathrm{S}\mathrm{O}$_0(n,1)$ in Sections 8, 9, and 10 to compute the relative Lie algebra cohomology groups $H^{\bullet} \big(\mathfrak{so}(n,1), $\mathrm{S}\mathrm{O}$(n); $\mathcal{P}$(V^k) \big)$\mathrm{.}Here$V = $\mathbb{R}$^{n,1}$isMinkowskispaceand$$\mathcal{P}$(V^k)$isthesubspaceof$L^2(V^k)$ 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), $\mathrm{S}\mathrm{O}$(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 $\mathcal{P}(V^k)\subset L^2(V^k)$.