  Let $G$ be a group with subgroup $H$, and let $(\pi,V)$ be a complex
representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$
on the space $\mathrm{Hom}_H(\pi,\mathbb{C})$ of $H$-invariant linear forms on
$V$, provides a representation $\chi_{\pi}$ of $N$ trivial on $H$, which is a
character when $\mathrm{Hom}_H(\pi,\mathbb{C})$ is one dimensional. If moreover
$G$ is a reductive group over a local field, and $\pi$ is smooth irreducible,
it is an interesting problem to express $\chi_{\pi}$ in terms of the possibly
conjectural Langlands parameter $\phi_\pi$ of $\pi$. In this paper we consider
the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division
algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$
is the centralizer of a non central element $\delta\in G$ such that $\delta^2$
is in the center of $G$, and $\pi$ has generic Jacquet-Langlands transfer to
$\mathrm{GL}_{md}(F)$. In this setting the space
$\mathrm{Hom}_H(\pi,\mathbb{C})$ is at most one dimensional. When
$\mathrm{Hom}_H(\pi,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that
the value of the $\chi_{\pi}$ on the non trivial class of $\frac{N}{H}$ is
$(-1)^m\epsilon(\phi_\pi)$ where $\epsilon(\phi_\pi)$ is the root number of
$\phi_{\pi}$. Along the way we extend many useful multiplicity one results for
linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we
also classify standard modules with linear periods and Shalika models, which
are new results even when $D=F$.
