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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
16/2020

Local cohomology on a subexceptional series of representations

András Cristian Lőrincz and Jerzy Weyman

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, \emph{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 $(A_5, \omega_3), \, (D_6, \omega_5)$ 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.

Received:
Jan 25, 2020
Published:
Jan 28, 2020

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Preprint
2020 Repository Open Access
András Christian Lőrincz and Jerzy Weyman

Local cohomology on a subexceptional series of representations