A Transverse Averaging Operator and Cohomology of Quotients by Homogeneous Non-closed Subgroups

In this article, we introduce a transverse averaging operator for basic forms on a complete Riemannian foliation with compact leaf closure space, equipped with an isometric transverse Lie algebra action. In contrast to the classical averaging operator in equivariant geometry, which is defined by integration over a compact Lie group, our operator is constructed purely from infinitesimal transverse data and does not require any global group action. We prove that every closed basic form is sent to an invariant basic form representing the same basic cohomology class.
The main application is formulated independently of foliation theory. We compute the diffeological de Rham cohomology of the homogeneous quotient $G/H$, where $G$ is a connected Lie group, not necessarily compact, and $H$ is a connected Lie subgroup, not necessarily closed. Let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, respectively. Assuming that $\mathfrak g$ is of compact type and that $G/\overline{H}$ is compact, we prove that $ H^\bullet_{\mathrm{dR}}(G/H) \cong H^\bullet(\mathfrak g,\mathfrak h)$. When $\mathfrak h$ is an ideal in $\mathfrak g$, the compact-type assumption on $\mathfrak g$ can be dropped, and under the sole assumption that $G/\overline{H}$ is compact we obtain $ H^\bullet_{\mathrm{dR}}(G/H) \cong H^\bullet(\mathfrak g/\mathfrak h)$. These results extend the classical Chevalley--Eilenberg computation from compact Lie groups and closed subgroups to homogeneous quotients $G/H$ by possibly non-closed subgroups $H$.