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EQUIVARIANT ${\mathcal{D}}$-MODULES ON ALTERNATING SENARY 3-TENSORS

Part of: (Co)homology theory General commutative ring theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  29 November 2019

ANDRÁS C. LŐRINCZ  and
MICHAEL PERLMAN
ANDRÁS C. LŐRINCZ
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, Leipzig, Germany04103 email lorincz@mis.mpg.de
MICHAEL PERLMAN
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN46556 email mperlman@nd.edu
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Abstract

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$. We describe explicitly the category of $\operatorname{GL}_{6}$-equivariant coherent ${\mathcal{D}}_{X}$-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$-modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 13D45: Local cohomology 13A50: Actions of groups on commutative rings; invariant theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 61 - 82
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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