Characters of equivariant 𝒟-modules on Veronese cones

Abstract

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