  We first consider the rational Cherednik algebra corresponding to the action
of a finite group on a complex variety, as defined by Etingof. We define a
category of representations of this algebra which is analogous to "category O"
for the rational Cherednik algebra of a vector space. We generalise to this
setting Bezrukavnikov and Etingof's results about the possible support sets of
such representations. Then we focus on the case of $S_n$ acting on $\C^n$,
determining which irreducible modules in this category have which support sets.
We also show that the category of representations with a given support, modulo
those with smaller support, is equivalent to the category of finite dimensional
representations of a certain Hecke algebra.
