Sur l'irr\'eductibilit\'e d'une induite parabolique

TL;DR

This paper investigates conditions under which certain induced representations over non-Archimedean fields have unique irreducible quotients, with explicit parameter computations relevant for Howe correspondence in dual pairs.

Contribution

It provides sufficient conditions for irreducibility of induced representations and computes Zelevinsky parameters in specific cases, advancing understanding of representation theory over division algebras.

Findings

Identifies conditions for unique irreducible quotients of induced representations.

Computes Zelevinsky parameters explicitly for cuspidal cases.

Facilitates explicit Howe correspondence for dual pairs of type II.

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.