  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.
