GENERIC IRREDUCIBILITY OF PARABOLIC INDUCTION FOR REAL REDUCTIVE GROUPS

Let G be a real reductive linear group in the Harish-Chandra class. Suppose that P is a parabolic subgroup of G with Lang-lands 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^GP (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^GP^(πM ⊠ χ_ν) is irreducible. Here the parameter ν is in a^∗ = (Lie(A) ⊗_R C)^∗and generic means outside a countable, locally finite union of hyperplanes which de-pends 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.