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On parabolic induction on inner forms of the general linear group over a non-archimedean local field. (English) Zbl 1355.22005

In the paper under the review, the authors study the irreducibility of the parabolic induction for the general linear group and its inner forms over a local non-archimedean field. Using algebraic methods, mostly based on the Geometric Lemma provided by Bernstein and Zelevinsky, enhanced by the intertwining operators method and the Zelevinsky involution, the authors provide a nice combinatorial criterion for the irreducibility of the induced representation when one of the representations which appear in the product is a segment representation.
The provided criterion is then directly extended to some other important classes of irreducible representations of the general linear group, such as unramified representations, generic representations and ladder representations. As a consequence, the authors also provide alternative and elegant proofs of some important results regarding the reducibility of the parabolic induction for the general linear group over a non-archimedean divison algebra.

MSC:

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