  We establish a transfer of unitarity for a Bernstein component of the
category of smooth representations of a reductive p-adic group to the
associated Hecke algebra, in the framework of the theory of types, whenever the
Hecke algebra is an affine Hecke algebra with geometric parameters, in the
sense of Lusztig (possibly extended by a group of automorphisms of the root
datum). It is known that there is a large class of such examples (detailed in
the paper). As a consequence, we establish relations between the unitary duals
of different groups, in the spirit of endoscopy.
