| The rational Cherednik algebra, defined by Etingof and Ginzburg, is an algebra associated to any finite group acting on a complex vector space. It arises as a double degeneration of the double affine Hecke algebra introduced by Cherednik. Thus it may be thought of as a member of the family of Hecke algebras associated to a Weyl group. It is also related to the Calogero-Moser integral system. Moreover it is a universal deformation of the algebra which acts on equivariant D -modules on the vector space.;We will study the representations of this algebra, focussing on type A. In doing so, we will be naturally led to consider the rational Cherednik algebra of a variety, rather than a vector space, as defined by Etingof. We define a category of representations of this algebra which is analogous to "category O " in the vector space setting. We generalise to this setting Bezrukavnikov and Etingof's results about the possible support sets of such representations. Then we focus on the type A case, that is, the symmetric group acting on Cn by permuting coordinates. We determine 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. |
