Neighborhoodsatinnityandthe方案.PDFVIP

Neighborhoods at infinity and the Plancherel formula for a reductive p-adic symmetric space 4 1 Patrick Delorme 0 2 abreviated title: Plancherel formula for a reductivep-adic symmetric r p space A 7 ] Aix Marseille Universit´e T CNRS-IML, FRE 3529 R. 163 Avenue de Luminy h 13288 Marseille Cedex 09 t a France. m delorme@iml.univ-mrs.fr [ 1 The author has been supported by the program ANR-BLAN08-2-326851 and by the v 0 Institut Universitaire de France during the elaboration of this work. 2 7 1 Abstract Yiannis Sakellaridis and Akshay Venkathesh have determined, when the 4. group G is split and the field F is of characteristic zero, the Plancherel formula for any 0 spherical space X for G modulo the knowledge of the discrete spectrum. 4 The starting point is the determination of good neighborhoods at infinity of X/J, 1 : where J is a small compact open subgroup of G. These neighborhoods are related to v i ”boundary degenerations” of X. The proof of their existence is made by using wonderful X compactifications. r a 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 formu