On dual groups of symmetric varieties and distinguished representations of $p$-adic groups

Abstract

Let $X=H\backslash G$ be a spherical 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 the little Weyl group of $X$. It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism $\widehat{\varphi}_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$. Conjecturally, this detects the $H$-distinguished representations of $G$. In this strictly utilitarian note, assuming $X$ is a symmetric variety, we give a more conceptual way of constructing the homomorphism $\widehat{\varphi}_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$, and make a few conjectures on how $\widehat{\varphi}_X$ is related to $H$-distinguished representations of $G$ by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of $G$ factors through $\widehat{\varphi}_X$ for any symmetric variety $X=H\backslash G$.