2006/12/19 by Patrick Delorme, Vincent Sécherre, Delorme, Patrick +2 · 1 citation
Mathematics · #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:22E50
paper · pdf · doi:10.48550/arxiv.math/0612545
33 pages
arxiv created 2006/12/19 · openalex publication_date 2006/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a non Archimedean locally compact field of residue characteristic different from 2, let G be a connected reductive group defined over F, let s be an involutive F-automorphism of G and H an open F-subgroup of the fixed points group of s. We denote by G(F) (resp. H(F)) the group of F-points of G (resp. H). In this paper, we obtain an analogue of the Cartan decomposition for the reductive symmetric space G(F)/H(F). More precisely, we obtain a decomposition of G(F) as a union of H(F)-cosets which is related to the H(F)-conjugacy classes of maximal s-anti-invariant F-split tori in G. When G is F-split, we get a more precise result, involving the stabilizer of a special point of the Bruhat-Tits building of G over F.