Equivariant $\mathcal {D}$-modules on $2\times 2\times n$ hypermatrices

Abstract:

We study $\mathcal {D}$-modules and related invariants on the space of $2\times 2\times n$ hypermatrices for $n\geq 3$, which has finitely many orbits under the action of $G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C})$. We describe the category of coherent $G$-equivariant $\mathcal {D}$-modules as the category of representations of a quiver with relations. We classify the simple equivariant $\mathcal {D}$-modules, determine their characteristic cycles and find special representations that appear in their $G$-structures. We determine the explicit $\mathcal {D}$-module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases $n=3$ and $n=4$, they are completely uniform for $n\geq 5$.