Local cohomology on a subexceptional series of representations
András Cristian Lőrincz and Jerzy Weyman
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Submission date: 25. Jan. 2020
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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 (C3,ω3),(A5,ω3),(D6,ω5) and (E7,ω6). In each of these four cases, the group G = G′× 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 DX-modules as the category of representations of a quiver with relations. We construct explicitly the simple G-equivariant DX-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 DX-module structure. As a consequence, we calculate the intersection cohomology groups and Lyubeznik numbers of the orbit closures. While our results for the cases (A5,ω3),(D6,ω5) and (E7,ω6) are still completely uniform, the case (C3,ω3) displays a surprisingly diﬀerent behavior. We give two explanations for this phenomenon: one topological, as the middle orbit of (C3,ω3) is not simply-connected; one geometric, as the closure of the orbit is not Gorenstein.