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 HomH(π,C) of H-invariant linear forms on V, provides a representation χπ of N trivial on H, which is a character when HomH(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 χπ in terms of the possibly conjectural Langlands parameter φ_π of π. In this paper we consider the following situation: G=GLₘ(D) for D a central division algebra of dimension d² over a p-adic field F, H is the centralizer of a non central element δ∈ G such that δ² is in the center of G, and π has generic Jacquet-Langlands transfer to GLmd(F). In this setting the space HomH(π,C) is at most one dimensional. When HomH(π,C) C and H≠ N, we prove that the value of the χπ on the non trivial class of N/H is (-1)ᵐε(φ_π) where ε(φ_π) is the root number of φπ. 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 $D=F$.