L’involution De Zelevinski Modulo

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