The sign of linear periods

Abstract

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 ${\varphi }_{\pi }$ 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 $H\ne N$, we prove that the value of the ${\chi }_{\pi }$ on the non trivial class of $\frac{N}{H}$ is $(-1{)}^m\epsilon ({\varphi }_{\pi })$ where $\epsilon ({\varphi }_{\pi })$ is the root number of ${\varphi }_{\pi }$. 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 $D=F$.

We extended the main result from $p$-adic to all local fields of characteristic zero in Section 7, thanks to the new Appendix D by M. Suzuki and H. Tamori which classifies Archimedean standard modules with a linear model