Rational Cherednik Algebras, Ep. 2: Attack of the finite modules

In the first part of the series (RCA's: The phantom bilinear form), we saw Rational Cherednik Algebras as a common generalization of the Weyl algebra and twisted group algebras. The very complicated structure of these objects makes it difficult to experiment with and further begs the question: Why would one care? We will try to address both issues. For one, we will discuss the most basic case, i.e. the cyclic group of order 2 (and if the speaker manages to figure it out in the next few hours, perhaps even the dihedral case). On the other hand, we will briefly present a very beautiful theory of finite dimensional simple modules of RCA's and consider their numerological and representation theoretic connections with popular combinatorial objects.