  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.
