Characters of equivariant $\mathcal{D}$-modules on Veronese cones

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