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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Characters of equivariant $\mathcal {D}$-modules on Veronese cones
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by Claudiu Raicu PDF
Trans. Amer. Math. Soc. 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.
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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