# Sur l&#x27;irréductibilité d&#x27;une induite parabolique - Mathematics > Representation Theory

Abstract: Let $F$ be a non-Archimedean locally compact field and let $D$ be a centraldivision algebra over $F$. Let $\pi 1$ and $\pi 2$ be respectively two smoothirreducible representations of ${ m GL}n 1,D$ and ${ m GL}n 2,F$, $n 1,n 2 \geq 0$. 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 ${ m GL}n 1+n 2,D$ has a unique irreducible quotient. In the casewhere $\pi 1$ is a cuspidal representation, we compute the Zelevinsky-sparameters of such a quotient in terms of parameters of $\pi 2$. This is thekey point for making explicit Howe correspondence for dual pairs of type II.

Author: Alberto Minguez LM-Orsay

Source: https://arxiv.org/