AbstractLetGbe a reductivep-adic group, we are interested in finitely generated projective smoothG-modules. LetPbe such a module, consider it as a З-module, where З is the Bernstein center of the category of smoothG-modules. Then we can formP⊗З.χℂ for every complex-valued character of З: it is a finite length smooth representation ofG. We describe its image in the Grothendieck group of finite length smoothG-modules. To do this, we define under suitable assumptions a З-valued character on the З-admissible (but not admissible!) representationP. The case of indGK(1) whereKis a special compact open subgroup ofGis an interesting example. Some of his properties are discussed and extended to other representations ofKusing Bushnell and Kutzko's theory of types, whenG= GL(n).