Rational Cherednik algebras were introduced by Etingof and
Ginzburg as a family of deformations of certain invariant rings related
to
complex reflection groups. These interesting algebras have subsequently
been shown to have very interesting properties and have many
applications
to representation theory, algebraic combinatroics and geometry.
We will focus on the representation theory of rational Cherednik
algebras
and in particular their "highest weight category" of modules, also known
as category O. It turns out that a crucial ingredient in the theory is
the
KZ functor from this category to the category of modules for a certain
cyclotomic Hecke algebra (at a root of unity). For wreath products of a
symmetric group and a cyclic group category O behaves in many ways like
the category of modules for cyclotomic q-Schur algebra: for instance,
the
KZ functor is much like the Schur functor. In fact it is a theorem of
Rouquier that for certain parameters of the rational Cherednik algebras
these two categories are equivalent. In general however, category O
provides a new analogue of the q-Schur algebra.
We shall explain in our lectures the above theory and, time permitting,
shall discuss the "t=0" theory which mirrors (at least conjecturally)
generic behaviour of Hecke algebras.