The representation theory of Hecke algebras of Weyl group is a classical topic in Lie
representation theory. Sometimes, the categories of modules have infinite homological
dimension (a.k.a. singular), for example, this happens when the parameters are roots
of unity of small enough order. Ginzburg, Guay, Opdam and Rouquier constructed
highest weight covers of the categories of modules over Hecke algebras (that should
be thought as resolutions of singularities). These covers are categories O over Rational
Cherednik algebras. In 2005, Rouquier has conjectured that all these covers are derived
equivalent (a result usually anticipated for nice resolutions in Algebraic geometry).
I will introduce the Rational Cherednik algebras, their categories O and the KZ functor.
Then I will sketch the proof of Rouquier's conjecture. Time permitting I will explain an
application to the computation of supports of the simple modules in categories O for
Weyl groups of classical types.