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Characters of equivariant
${\mathcal{D}}$ -modules on spaces of matrices
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 9 / September 2016
- Published online by Cambridge University Press:
- 28 June 2016, pp. 1935-1965
- Print publication:
- September 2016
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- Article
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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.
