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On the irreducibility of an induced parabolic. (Sur l’irréductibilité d’une induite parabolique.) (French) Zbl 1172.22008
Let $$D$$ be a central division algebra over a non-archimedean local field. When $$\pi_1$$ and $$\pi_2$$ are two irreducible smooth representations of GL$$(n_1,D)$$ and GL$$(n_2,D)$$ respectively, we have the parabolically induced representation $$\pi_1\times\pi_2$$ of GL$$(n_1+ n_2,D)$$. The question dealt with in the article is: under what conditions on $$\pi_1$$ and $$\pi_2$$ does $$\pi_1\times\pi_2$$ have a unique irreducible quotient (resp. a unique irreducible submodule)? The classification by Tadić of the irreducible representations of GL$$(n,D)$$ is used to formulate and prove a sufficient condition on $$\pi_2$$ such that, for any irreducible representation $$\pi_1$$, $$\pi_1\times\pi_2$$ has a unique irreducible submodule. The author also proves and applies another classification of the representations of GL$$(n,D)$$, where submodules are involved instead of quotients as in Tadić’s classification. An important tool is Zelevinsky’s combinatorial lemma, which gives a composition sequence for the parabolic restriction of a parabolically induced representation. The parameters of the unique irreducible quotient of $$\pi_1\times\pi_2$$ with cuspidal $$\pi_1$$ are explicitly computed in terms of the parameters of $$\pi_2$$. This computation is useful for the explicit computation of a theta correspondence, cf. A. Mínguez [Ann. Sci. Éc. Norm. Supér., VI, 41, No. 5, 717–741 (2008; Zbl 1220.22014)].

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields