Characters of equivariant D-modules on Veronese cones

TL;DR

This paper characterizes simple equivariant D-modules on Veronese cones, provides a counterexample to a conjecture for d=2, and analyzes local cohomology modules, revealing their simplicity and unique non-zero instance.

Contribution

It describes characters of simple equivariant D-modules on Veronese cones and presents a counterexample to Levasseur's conjecture for d=2, advancing understanding of D-module structures.

Findings

Characterization of simple equivariant D-modules on Veronese cones.

Counterexample to Levasseur's conjecture for d=2.

Identification of the unique non-zero local cohomology module as simple.

Abstract

For d > 1, we consider the Veronese map of degree d on a complex vector space W , Ver_d : W -> Sym^d W , w -> w^d , and denote its image by Z. We describe the characters of the simple GL(W)-equivariant holonomic D-modules supported on Z. In the case when d is 2, we obtain a counterexample to a conjecture of Levasseur by exhibiting a GL(W)-equivariant D-module on the Capelli type representation Sym^2 W which contains no SL(W)-invariant sections. We also study the local cohomology modules H_Z^j(S), where S is the ring of polynomial functions on the vector space Sym^d W. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree j = codim(Z)), and moreover we prove that it is a simple D-module and determine its character.