Original scientific paper

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

Generic irreducibility of parabolic induction for real reductive groups

David Renard ; Centre de mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau Cedex, France

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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.

Keywords

Representation of real reductive groups, generic irreducibility of parabolic induction, Kazhdan-Lusztig-Vogan algorithm

Hrčak ID:

325174

URI

https://hrcak.srce.hr/325174

Publication date:

27.12.2024.

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