Generic irreducibility of parabolic induction for real reductive groups

2023/10/17 by David Renard, Renard, David
Mathematics · #22E45 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.11202

openalex publication_date 2023/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 π be an irreducible representation of the Levi factor L=MA. We give sufficient conditions on the infinitesimal character of π for the induced representation i_P^G(π) to be irreducible. In particular, we prove that if π_M is an irreducible representation of M, then for a generic character χ_ν of A, the induced representation i_P^G(π_M\boxtimes χ_ν) is irreducible. Here the parameter ν is in \mathfraka^*=(Lie(A)⊗_\mathbb R \mathbb C)^* and generic means outside a countable, locally finite union of hyperplanes which depends only on the infinitesimal character of π. Notice that there is no other assumption on π or π_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.

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