Unitary representations of cyclotomic rational Cherednik algebras

Abstract

We classify the irreducible unitary modules in category Oc 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 λ• of n, the closed semi-linear set of parameters c for which the contravariant form on the irreducible representation Lc(λ•) 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.