The sign of linear periods

Let G be a group with subgroup H, and let (π,V) be a complex representation of G. The natural action of the normalizer N of H in G on the space Hom_H(π,ℂ) of H-invariant linear forms on V, provides a representation χ_π of N trivial on H, which is a character when Hom_H(π,ℂ) is one dimensional. If moreover G is a reductive group over a p-adic field, and π is smooth irreducible, it is an interesting problem to express χ_π in terms of the possibly conjectural Langlands parameter ϕ_π of π. In this paper we consider the following situation: G=GL_m(D) for D a central division algebra of dimension d^2 over a p-adic field F, H is the centralizer of a non central element δ∈ G such that δ^2 is in the center of G, and π has generic Jacquet-Langlands transfer to GL_md(F). In this setting the space Hom_H(π,ℂ) is at most one dimensional. When Hom_H(π,ℂ)≃ℂ and H≠ N, we prove that the value of the χ_π on the non trivial class of N/H is (-1)^mϵ(ϕ_π) where ϵ(ϕ_π) is the root number of ϕ_π. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split G, and we also classify standard modules with linear periods and Shalika models, which are new results even when D=F.