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.
Categories: math.RT