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Characters of equivariant ${\mathcal{D}}$ -modules on spaces of matrices

Part of: Special varieties Commutative algebra: Homological methods (Co)homology theory

Published online by Cambridge University Press:  28 June 2016

Claudiu Raicu
Claudiu Raicu*
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA email craicu@nd.edu Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, Sector 1, 21 Calea Grivitei str., PO Box 1-764, Bucharest, Romania
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Abstract

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We compute the characters of the simple $\text{GL}$ -equivariant holonomic ${\mathcal{D}}$ -modules on the vector spaces of general, symmetric, and skew-symmetric matrices. We realize some of these ${\mathcal{D}}$ -modules explicitly as subquotients in the pole order filtration associated to the $\text{determinant}/\text{Pfaffian}$ of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the ${\mathcal{D}}$ -module composition factors of local cohomology modules with determinantal and Pfaffian support.

Keywords

equivariant D-modulesrank stratificationslocal cohomology

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 13D45: Local cohomology 14M12: Determinantal varieties
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1935 - 1965
Copyright
© The Author 2016 

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