D-modules and Solvable Lie Foliations

Abstract

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the ${\mathcal{D}}_X$-module ${\mathcal{M}}_{\mathcal{I}}={\mathcal{D}}_X/{\mathcal{D}}_X\cdot \mathcal{I}$ and the derived ring ${\mathcal{D}}_{\mathcal{I}}:=R\mathcal{H}o{m}_{{\mathcal{D}}_X}({\mathcal{M}}_{\mathcal{I}},{\mathcal{M}}_{\mathcal{I}})$. We compute the D-irregularity $\mathrm{D}\text{-}\mathrm{irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of ${\mathcal{D}}_{\mathcal{I}}$, establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.