## On Harish-Chandra bimodules of rational Cherednik algebras at regular parameter values

##### Abstract

This thesis deals with Harish-Chandra bimodules of rational Cherednik algebras Hk at regular
parameter values k, that is those k for which Hk is a simple algebra. Rational Cherednik
algebras can be associated to any re
ection representation of a complex re
ection group .
The second chapter presents a review of some important results regarding rational Cherednik
algebras and their category Ok, which will be frequently used throughout.
The third chapter contains basic results about Harish-Chandra bimodules and the structure
of the category HCk of Harish-Chandra bimodules, many of which are new in the context
of complex re
ection groups but have known analogues for real re
ection groups at integral
parameter values by work of Berest-Etingof-Ginzburg in [BEG03b]. In particular we show that
if k is regular, then HCk is a semisimple tensor category and is equivalent to a tensor-closed
subcategory of modules of the associated Hecke algebra. Using work of I. Losev in [Los11a],
we also deduce that HCk is equivalent as a tensor category to repC (=Nk), the representation
category of a quotient of the complex re
ection group . This extends previous results for the
case of integral k. We manage to obtain some numerical consequences for the presentation of
the Hecke algebra of , which is linked to Hk and Ok via the KZk-functor.
The fourth chapter again is a review of standard results on Morita equivalences between rings
and integral shift functors giving Morita equivalences between rational Cherednik algebras and
their tensor categories of Harish-Chandra bimodules at di erent regular parameter values. The
case of integral parameter values k is discussed brie
y, going back to Berest-Etingof-Ginzburg
in [BEG03b] and Berest-Chalykh in [BC09]. The fth chapter gives a complete description of
Nk and its dependence on k (still for k regular) for the case that is cyclic.
In chapter 6 we deal with nite-dimensional Harish-Chandra bimodules of rational Cherednik
algebras associated to cyclic groups, compute the quiver of that category and derive a criterion
for wildness of HCk in the cyclic case.
Chapter 7 nally extends a classi cation of the structure of HCk as a tensor category for regular
k to nite irreducible Coxeter groups.