English

Generic irreducibility of parabolic induction for real reductive groups

Representation Theory 2024-07-12 v4

Abstract

Let GG be a real reductive linear group in the Harish-Chandra class. Suppose that PP is a parabolic subgroup of GG with Langlands decomposition P=MANP=MAN. Let π\pi be an irreducible representation of the Levi factor L=MAL=MA. We give sufficient conditions on the infinitesimal character of π\pi for the induced representation iPG(π)i_P^G(\pi) to be irreducible. In particular, we prove that if πM\pi_M is an irreducible representation of MM, then for a generic character χν\chi_\nu of AA, the induced representation iPG(πMχν)i_P^G(\pi_M\boxtimes \chi_\nu) is irreducible. Here the parameter ν\nu is in a=(Lie(A)RC)\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 π\pi. Notice that there is no other assumption on π\pi or πM\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

Cite

@article{arxiv.2310.11202,
  title  = {Generic irreducibility of parabolic induction for real reductive groups},
  author = {David Renard},
  journal= {arXiv preprint arXiv:2310.11202},
  year   = {2024}
}

Comments

18 pages, version 3, typos, mistakes and omissions corrected, references added, statements of the main results reformulated

R2 v1 2026-06-28T12:53:15.594Z