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 $\hat{G}$ be the Langlands dual group of $G$. There is a complex group ${\hat{G}}_X$ 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 ${\hat{\phi }}_X:{\hat{G}}_X\times {\mathrm{SL}}_2(\mathbb{C})\to \hat{G}$. 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 ${\hat{\phi }}_X:{\hat{G}}_X\times {\mathrm{SL}}_2(\mathbb{C})\to \hat{G}$, and make a few conjectures on how ${\hat{\phi }}_X$ 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 ${\hat{\phi }}_X$ for any symmetric variety $X=H\backslash G$.

Slightly revised