Neighborhoods at infinity and the Plancherel formula for a reductive $p$-adic symmetric space

Abstract

Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group $G$ is split and the field $\F$ is of characteristic zero, the Plancherel formula for any spherical space $X$ for $G$ modulo the knowledge of the discrete spectrum. The starting point is the determination of good neighborhoods at infinity of $X/J$, where $J$ is a small compact open subgroup of $G$. These neighborhoods are related to "boundary degenerations" of $X$. The proof of their existence is made by using wonderful compactifications. In this article we will show the existence of such neighborhoods assuming that $\F$ is of characteristic different from 2 and $X$ is symmetric. In particular, one does not assume that $G$ is split. Our main tools are the Cartan decomposition of Benoist and Oh, our previous definition of the constant term and asymptotic properties of Eisenstein integrals due to Nathalie Lagier . Once the existence of these neighborhoods at infinity of $X$ is established, the analog of the work of Sakellaridis and Venkatesh is straightforward and leads to the Plancherel formula for $X$.