Sur l'irr\'eductibilit\'e 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 ${\rm GL}(n_1,D)$ and ${\rm GL}(n_2,F)$, $n_1, n_2 \geq 0$. 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 ${\rm GL}(n_1+n_2,D)$ 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.