The $\ell$-modular Zelevinski involution

Abstract

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, 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 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.