Symmetry often turns complicated differential equations into structured and computable objects. In the language of algebraic analysis, this principle gives rise to a rich class of equivariant D-modules associated with group actions. Important examples include GKZ hypergeometric systems and tautological systems attached to homogeneous varieties such as Grassmannians and more general flag varieties.

In this talk, I will present recent results showing that many of these symmetry-driven D-modules possess a deeper geometric meaning: they admit a natural mixed Hodge module interpretation and therefore encode topological information about the underlying spaces. This perspective not only clarifies the structure of their solution spaces but also suggests new tools for their study.

I will explain how Lie algebroid methods provide a natural framework for understanding D-modules arising from group actions, revealing connections between symmetry, geometry, topology, and differential equations. In particular, they offer new tools for studying the duality theory of tautological systems and their Hodge-theoretic properties.

Finally, I will discuss ongoing work on toric degenerations, which aims to connect highly symmetric geometric situations with more combinatorial models.

This talk is based on joint works with Thomas Reichelt, Christian Sevenheck, Avi Steiner and Uli Walther.