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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)].

22E50 Representations of Lie and linear algebraic groups over local fields