Characters of equivariant -modules on spaces of matrices

Abstract

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.