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

Published online by Cambridge University Press:  28 June 2016

Claudiu Raicu*
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA email 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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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.

Research Article
© The Author 2016 


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