  We prove that for every Bushnell-Kutzko type that satisfies a certain
rigidity assumption, the equivalence of categories between the corresponding
Bernstein component and the category of modules for the Hecke algebra of the
type induces a bijection between irreducible unitary representations in the two
categories. This is a generalization of the unitarity criterion of Barbasch and
Moy for representations with Iwahori fixed vectors.
