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

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**369**(2017), 2087-2108 Request permission

## 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.## References

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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
- MR Author ID: 909516
- Email: craicu@nd.edu
- Received by editor(s): December 28, 2014
- Received by editor(s) in revised form: April 4, 2015
- Published electronically: May 3, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 2087-2108 - MSC (2010): Primary 13D45, 14M17, 14F10, 14F40
- DOI: https://doi.org/10.1090/tran/6713
- MathSciNet review: 3581228