Local cohomology on a subexceptional series of representations

We consider a series of four subexceptional representations com-ing from the third line of the Freudenthal-Tits magic square; using Bourbaki nota-tion, these are representations (G ', X) corresponding to (C3, w3), (A5, w3), (D6, w5), and (E7, w6). In each case X has five G = G ' xC-orbits, displaying some uniform behavior, e.g. their dimensions or defining ideals. In this paper, we determine some further invariants and analyze their uniformity within the series. We describe the category of G-equivariant coherent DX-modules as the category of representations of a quiver. We construct explicitly the simple equivariant D-modules and describe their G-structures. We determine the D-module structure of local cohomology mod-ules supported in orbit closures, and calculate intersection cohomology groups and Lyubeznik numbers. While our results for (A5, w3), (D6, w5), (E7, w6) are still com-pletely uniform, the case (C3, w3) displays a surprisingly different behavior, for which we give two explanations: the middle orbit is not simply-connected, and its closure is not Gorenstein.