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

Abstract

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