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dc.contributor.advisorGordon, Iain
dc.contributor.authorBellamy, Gwyn
dc.date.accessioned2011-02-01T10:09:38Z
dc.date.available2011-02-01T10:09:38Z
dc.date.issued2010
dc.identifier.urihttp://hdl.handle.net/1842/4733
dc.description.abstractThe 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.en
dc.language.isoenen
dc.publisherThe University of Edinburghen
dc.relation.hasversionG. Bellamy. The Calogero-Moser partition for G(m; d; n). arXiv, 0911.0066v1, 2009.en
dc.relation.hasversionG. Bellamy. Cuspidal representations of rational Cherednik algebras at t = 0. arXiv, 0911.0069v1, 2009en
dc.relation.hasversionG. Bellamy. Factorization in generalized Calogero-Moser spaces. J. Algebra, 321(1):338{344, 2009.en
dc.relation.hasversionG. Bellamy. On singular Calogero-Moser spaces. Bull. Lond. Math. Soc., 41(2):315{326, 2009.en
dc.subjectrepresentation theoryen
dc.titleGeneralized Calogero-Moser spaces and rational Cherednik algebrasen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen


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