Local cohomology on a subexceptional series of representations

András Cristian Lőrincz and Jerzy Weyman

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Submission date: 25. Jan. 2020
Pages: 23
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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 (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 D_X-modules as the category of representations of a quiver with relations. We construct explicitly the simple G-equivariant 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 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 (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.