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 p1 and p2 be respectively two smooth irreducible representations of GLðn1; DÞ and GLðn2; DÞ, n1; n2 f0. In this article, we give some su‰cient conditions on p1 and p2 so that the parabolically induced representation of p1 np2 to GLðn1 þ n2; DÞ has a unique irreducible quotient. In the case where p1 is a cuspidal representation, we compute the Zelevinsky’s parameters of such a quotient in terms of the parameters of p2. This is the key point for making explicit Howe correspondence for dual pairs of type II (cf. [A. Mínguez, Correspondance de Howe explicite : paires duales de type II, Ann. Ec. Norm. Sup., à paraître]).