Local cohomology on a subexceptional series of representations

Abstract

We consider a series of four subexceptional representations coming from the third line of the Freudenthal-Tits magic square; using Bourbaki notation, these are fundamental representations $(G',X)$ corresponding to $(C_3, \omega_3),\, (A_5, \omega_3), \, (D_6, \omega_5)$ and $(E_7, \omega_6)$. In each of these four cases, the group $G=G'\times \mathbb{C}^*$ acts on $X$ with five orbits, and many invariants display a uniform behavior, e.g. dimension of orbits, their defining ideals and the character of their coordinate rings as $G$-modules. In this paper, we determine some more subtle invariants and analyze their uniformity within the series. We describe the category of $G$-equivariant coherent $\mathcal{D}_X$-modules as the category of representations of a quiver with relations. We construct explicitly the simple $G$-equivariant $\mathcal{D}_X$-modules and compute the characters of their underlying $G$-structures. We determine the local cohomology groups with supports given by orbit closures, determining their precise $\mathcal{D}_X$-module structure. As a consequence, we calculate the intersection cohomology groups and Lyubeznik numbers of the orbit closures. While our results for the cases $(A_5, \omega_3), \, (D_6, \omega_5)$ and $(E_7, \omega_6)$ are still completely uniform, the case $(C_3, \omega_3)$ displays a surprisingly different behavior. We give two explanations for this phenomenon: one topological, as the middle orbit of $(C_3, \omega_3)$ is not simply-connected; one geometric, as the closure of the orbit is not Gorenstein.