Three problems in the representation theory of rational Cherednik algebras

The rational Cherednik algebra is an algebra attached to a complex reflection group W and depending on a set of central parameters indexed by the conjugacy classes of reflections in W. Its representation theory is roughly analogous to that of a simple Lie algebra, but many important questions remain unresolved.

The eponymous problems are to classify the unitary irreducible representations, classify the finite dimensional irreducible representations, and to construct canonical bases for the W-invariant subspace in a given irreducible module.

For the symmetric group, Berest-Etingof-Ginzburg classified the finite dimensional irreducible modules, and Etingof-Stoica made a conjectural classification of the unitary irreducibles. As it turns out, their conjecture follows from work of Suzuki, and a similar technique ought to classify the unitary irreducibles for all the groups in the infinite family G(r,p,n). Though we have many examples of finite dimensional representations for the rational Cherednik algebra of type G(r,p,n), there is not even a conjectural classification available in general.

The last problem, giving canonical bases for the W-invariant subspace, is closely related the theory of non-crossing partitions.