Characters of equivariant D-modules on spaces of matrices

TL;DR

This paper computes characters of simple GL-equivariant holonomic D-modules on matrix spaces, explicitly realizing some as subquotients and others as local cohomology modules, and addresses a conjecture with counterexamples.

Contribution

It provides explicit character formulas for equivariant D-modules on matrices and proves a conjecture of Levasseur, with new counterexamples for symmetric matrices.

Findings

Computed characters of simple equivariant D-modules on matrices.

Realized some D-modules as subquotients in pole order filtrations.

Provided counterexamples to Levasseur's conjecture for symmetric matrices.

Abstract

We compute the characters of the simple GL-equivariant holonomic D-modules on the vector spaces of general, symmetric and skew-symmetric matrices. We realize some of these D-modules explicitly as subquotients in the pole order filtration associated to the determinant/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 D-module composition factors of local cohomology modules with determinantal and Pfaffian support.