Repr\'esentations et quasi-caract\`eres de niveau 0; endoscopie

Abstract

Let F be a finite extension of Q_p and let G be a connected reductive group over F. We assume that p is big relatively to G. Let G' be an endoscopic group of G. Following Arthur, we have, roughly speaking, a spectral transfer which, to a stable finite linear combination of irreducible admissible representations of G'(F), associates a finite linear combination of irreducible admissible representations of G(F). Let p^{0,G} be the Bernstein's projector such that, for an irreducible admissible representation $\pi$ of G(F), we have p^{0,G}($\pi$)=$\pi$ if $\pi$ has level 0 and p^{0,G}($\pi$)=0 if $\pi$ has strictly positive level. Define similarly p^{0,G'}. We prove that p^{0,G'} preserves the space of stable finite linear combination of irreducible admissible representations of G'(F) and that p^{0,G} transfer=transfer p^{0,G'}.