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Characters of equivariant $ \mathcal{D}$-modules on Veronese cones

Author: Claudiu Raicu
Journal: Trans. Amer. Math. Soc. 369 (2017), 2087-2108
MSC (2010): Primary 13D45, 14M17, 14F10, 14F40
Published electronically: May 3, 2016
MathSciNet review: 3581228
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Abstract: For $ d>1$, we consider the Veronese map of degree $ d$ on a complex vector space $ W$, $ \mathrm {Ver}_d:W\longrightarrow \mathrm {Sym}^d W$, $ w\mapsto w^d$, and denote its image by $ Z$. We describe the characters of the simple $ \mathrm {GL}(W)$-equivariant holonomic $ \mathcal {D}$-modules supported on $ Z$. In the case when $ d=2$, we obtain a counterexample to a conjecture of Levasseur by exhibiting a $ \mathrm {GL}(W)$-equivariant $ \mathcal {D}$-module on the Capelli type representation $ \mathrm {Sym}^2 W$ which contains no $ \mathrm {SL}(W)$-invariant sections. We also study the local cohomology modules $ H^{\bullet }_Z(S)$, where $ S$ is the ring of polynomial functions on the vector space $ \mathrm {Sym}^d W$. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree $ \bullet =\textrm {codim}(Z)$), and moreover we prove that it is a simple $ \mathcal {D}$-module and determine its character.

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Additional Information

Claudiu Raicu
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, Indiana 46556 – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania

Keywords: $\mathcal{D}$-modules, Veronese cones, local cohomology
Received by editor(s): December 28, 2014
Received by editor(s) in revised form: April 4, 2015
Published electronically: May 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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