Unitary representations of cyclotomic rational Cherednik algebras

Abstract

We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm determining, for each r-partition of n, the closed semi-linear set of parameters for which the contravariant form on the irreducible representation with the given r-partition as lowest weight is positive definite. We use this algorithm to give a closed form answer for the Cherednik algebra of the symmetric group (recovering a result of Etingof-Stoica and the author) and the Weyl groups of classical type.