Sur l'irréductibilité d'une induite parabolique

Abstract

Let $F$ be a non-Archimedean locally compact field and let $D$ be a central division algebra over $F$. Let ${\pi }_1$ and ${\pi }_2$ be respectively two smooth irreducible representations of $\mathrm{GL}(n_1,D)$ and $\mathrm{GL}(n_2,F)$, $n_1,n_2\ge 0$. In this article, we give some sufficient conditions on ${\pi }_1$ and ${\pi }_2$ so that the parabolically induced representation of ${\pi }_1\otimes {\pi }_2$ to $\mathrm{GL}(n_1+n_2,D)$ has a unique irreducible quotient. In the case where ${\pi }_1$ is a cuspidal representation, we compute the Zelevinsky's parameters of such a quotient in terms of parameters of ${\pi }_2$. This is the key point for making explicit Howe correspondence for dual pairs of type II.