A D-module approach to invariant distributions with finitely many orbits

Abstract

Tauchi provides an example illustrating the action of a real algebraic subgroup $H$ of $GL(2n, \mathbb{R})$ with finitely many orbits on $\mathbb{R}^{2n}$, while the dimension of the space of relative $H$-invariant distributions on $\mathbb{R}^{2n}$ is infinite. We offer a perspective on this example from the viewpoint of D-modules, where we explicitly determine the simple quotient regular holonomic D-modules and demonstrate that the distributions exhibit an enlarged symmetry.