Representations of the symplectic reflection algebras and of the rational Cherednik algebras.

摘要

Given a symplectic reflection group W one can define a symplectic reflection algebra Hk( W). When W is a complex reflection group, the corresponding algebra is a called rational Cherednik algebra. The representation theory of these algebras is similar to the representation theory of Lie algebras. In particular, one can define the category O of Hk(W)-modules.; In the thesis we study the rational Cherednik algebras associated to asymmetric group, to a group of the form Sn imes;Z/lZ n and to a dihedral group. For a symmetric group and for S n imes;Z/lZ n we study the structure of the standard module corresponding to the trivial representation, while for a dihedral group we explicitly describe the structure of all the standard modules. In particular, we compute their Jordan-Holder series and the characters of the corresponding irreducible modules. We use symmetry arguments, the KZ-functor, and the results of C. Dunkl, E. Opdam, and M. F. E. De Jeu concerning singular vectors in the polynomial representation of Hk( W).; We also introduce the twisted rational Cherednik algebras, which arise naturally when considering the rational Cherednik algebras H k(W, V) associated to a non-faithful representation G → GL(V). In this case, the category of Hk(G, V)-modules splits into a direct sum of subcategories, each equivalent to the category of modules over a twisted rational Cherednik algebra associated to a certain subgroup of W acting on V. To prove this we use the theory of projective representations of finite groups and the results of V. Ostrik concerning module categories over the category of finite dimensional representations of a finite group.