Time: May 15, Wed., 16:00-17:00

Venue: Lecture Hall

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 $\Hom_H(\pi,\BC)$ of $H$-invariant linear forms on $V$, provides a representation $\chi_{\pi}$ of $ \frac{N}{H} $, which is a character when $\Hom_H(\pi,\BC)$ is one dimensional. If moreover $G$ is a reductive group over a $p$-adic 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 $\pi$. We will 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 $\delta\in G$ such that $\delta^2$ is in the center of $G$, and $\pi$ has generic Jacquet-Langlands transfer to $GL_{md}(F)$. In this setting the space $\Hom_H(\pi,\mathbb{C})$ is at most one dimensional. When $\Hom_H(\pi, \mathbb{C} )\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $\chi_{\pi}$ on the non trivial class of $\frac{N}{H}$ is $(-1)^m\e(\phi_\pi)$ where $\e(\phi_\pi)$ is the root number of $\phi_{\pi}$. This is a joint work with U.K. Anandavardhanan, H. Lu, V.  Sécherre  and C. Yang.