L’involution de Zelevinski modulo ℓ

Let F \mathrm {F} be a non-Archimedean locally compact field with residual characteristic p p , let G \mathrm {G} be an inner form of G L n ( F ) \mathrm {GL}_n(\mathrm {F}) , n ⩾ 1 n\geqslant 1 and let R \mathrm {R} be an algebraically closed field of characteristic different from p p . When R \mathrm {R} has characteristic ℓ > 0 \ell >0 , the image of an irreducible smooth R \mathrm {R} -representation π \pi of G \mathrm {G} by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G \mathrm {G} ) contains a unique irreducible term π ⋆ \pi ^\star with the same cuspidal support as π \pi . This defines an involution π ↦ π ⋆ \pi \mapsto \pi ^\star on the set of isomorphism classes of irreducible R \mathrm {R} -representations of G \mathrm {G} , that coincides with the Zelevinski involution when R \mathrm {R} is the field of complex numbers. The method we use also works for F \mathrm {F} a finite field of characteristic p p , in which case we get a similar result.