CHARACTERS OF EQUIVARIANT D-MODULES ON VERONESE CONES

摘要

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 = 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 center dot(S), where S is the ring of polynomial functions on the vector space SymdW. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree center dot = codim(Z)), and moreover we prove that it is a simple D-module and determine its character.