L’involution de Zelevinski modulo $\ell$

Abstract:

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