Equivariant D-modules on binary cubic forms

2017/12/28 by András C. Lőrincz, Claudiu Raicu, Lőrincz, András C. +3
Mathematics · #13D45 #14F10 #16G20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #math.AC #math.AG #math.RT #msc:13D45 #msc:14F10 #msc:16G20

paper · pdf · doi:10.48550/arxiv.1712.09932

arxiv created 2017/12/28 · arxiv updated 2017/12/29

Abstract

We consider the space X = Sym³(C²) of binary cubic forms, equipped with the natural action of the group GL₂ of invertible linear transformations of C². We describe explicitly the category of GL₂-equivariant coherent D_X-modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant D_X-modules (of which there are 14), and give formulas for the characters of their underlying GL₂-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.

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