Year: 2024
Type: article
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.
Source: Glasnik Matematicki
DOI: 10.3336/gm.59.2.05
Author: David Renard
Institution: École Polytechnique
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4405820524