Generic irreducibility of parabolic induction for real reductive groups

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.