Generalized Calogero-Moser spaces and rational Cherednik algebras

Abstract

The subject of this thesis is the interplay between the geometry and the representation theory of rational Cherednik algebras at t = 0. Exploiting this relationship, we use representation theoretic techniques to classify all complex re ection groups for which the geometric space associated to a rational Cherednik algebra, the generalized Calogero-Moser space, is singular. Applying results of Ginzburg-Kaledin and Namikawa, this classification allows us to deduce a (nearly complete) classification of those symplectic reflection groups for which there exist crepant resolutions of the corresponding symplectic quotient singularity. Then we explore a particular way of relating the representation theory and geometry of a rational Cherednik algebra associated to a group W to the representation theory and geometry of a rational Cherednik algebra associated to a parabolic subgroup of W. The key result that makes this construction possible is a recent result of Bezrukavnikov and Etingof on completions of rational Cherednik algebras. This leads to the definition of cuspidal representations and we show that it is possible to reduce the problem of studying all the simple modules of the rational Cherednik algebra to the study of these nitely many cuspidal modules. We also look at how the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space can be "factored" in terms of parabolic subgroups when it is restricted to particular subvarieties. In particular, we are able to confirm a conjecture of Etingof and Ginzburg on "factorizations" of the Etingof-Ginzburg sheaf. Finally, we use Clifford theoretic techniques to show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m; d; n) from the corresponding partition for G(m; 1; n). This confirms, in the case W = G(m; d; n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.