Representations of reductive groups distinguished by symmetric subgroups

Abstract: Let $G$ be a complex reductive group and $H=G^{\theta}$ be its fixed point subgroup under a Galois involution $\theta$. We show that any $H$-distinguished representation $\pi$ (i.e $\mathrm{dim}{\mathbb{C}}\left(\pi^{*}\right)^{H}\neq0$) satisfies: 1) $\pi^{\theta}\simeq\tilde{\pi}$, where $\tilde{\pi}$ is the contragredient representation and $\pi^{\theta}$ is the twist of $\pi$ under $\theta$. 2) $\mathrm{dim}{\mathbb{C}}\left(\pi^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.