  Let $X=H\backslash G$ be a symmetric variety over a $p$-adic field. Assume
$G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a
complex group $\widehat{G}_X$ whose root datum is naturally constructed from
that of $\widehat{G}$. In this paper, we construct a homomorphism
$\widehat{\varphi}_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to
\widehat{G}$ naturally and somewhat explicitly, and make a few conjectures on
how $\widehat{\varphi}_X$ is related to $H$-distinguished representations of
$G$. We will also show that the local Langlands parameter of the trivial
representation of $G$ factors through $\widehat{\varphi}_X$ for any symmetric
variety $X=H\backslash G$.
  Our group $\widehat{G}_X$ is different from the dual group by
Sakellaridis-Venkatesh. However, we will show that our conjectures are
consistent with various known examples and conjectures, especially in the
framework of the theory of Kato-Takano on relative cuspidality and relative
square integrability.
