Glasnik Matematicki, Vol. 59, No. 2 (2024), 327-349. \( \)

GENERIC IRREDUCIBILITY OF PARABOLIC INDUCTION FOR REAL REDUCTIVE GROUPS

David Renard

Centre de mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau Cedex, France
e-mail:david.renard@polytechnique.edu

Abstract.   Let \(G\) be a real reductive linear group in the Harish-Chandra class. Suppose that \(P\) is a parabolic subgroup of \(G\) with Langlands decomposition \(P=MAN\). Let \(\pi\) be an irreducible representation of the Levi factor \(L=MA\). We give sufficient conditions on the infinitesimal character of \(\pi\) for the induced representation \(i_P^G(\pi)\) to be irreducible. In particular, we prove that if \(\pi_M\) is an irreducible representation of \(M\), then, for a generic character \(\chi_\nu\) of \(A\), the induced representation \(i_P^G(\pi_M\boxtimes \chi_\nu)\) is irreducible. Here the parameter \(\nu\) is in \(\mathfrak{a}^*=(\mathrm{Lie}(A)\otimes_{\mathbb{R}} {\mathbb{C}})^*\) and generic means outside a countable, locally finite union of hyperplanes which depends only on the infinitesimal character of \(\pi\). Notice that there is no other assumption on \(\pi\) or \(\pi_M\) than being irreducible, so the result is not limited to generalised principal series or standard representations, for which the result is already well known.

2020 Mathematics Subject Classification.   20G05, 22E45

Key words and phrases.   Representation of real reductive groups, generic irreducibility of parabolic induction, Kazhdan-Lusztig-Vogan algorithm

https://doi.org/10.3336/gm.59.2.05

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