engleski

On reducibility of parabolic induction

Jacquet modules of a reducible parabolically induced representation of a reductive p-adic group reduce in a way consistent with the transitivity of Jacquet modules. This fact can be used for proving irreducibility of parabolically induced representations. Classical groups are particularly convenient for application of this method, since we have a very good information about part of the representation theory of their Levi subgroups (general linear groups are factors of Levi subgroups, and therefore we can apply the Bernstein-Zelevinsky theory). In the paper, we apply this type of approach to the problem of determining reducibility of parabolically induced representations of p-adic Sp(n) and SO(2n+1). We present also a method for getting Langlands parameters of irreducible subquotients. In general, we describe reducibility of certain generalized principal series (and some other interesting parabolically induced representations) in terms of the reducibility in the cuspidal case. When the cuspidal reducibility is known, we get explicit answers (for example, for representations supported in the minimal parabolic subgroups, the cuspidal reducibility is well-known rank one reducibility).

parabolic induction; Jacquet modules; irreducible representations; Jordan-Hoelder series

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