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An analogue of the Cartan decomposition for p-adic reductive symmetric spaces

Patrick Delorme·Vincent Secherre
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Abstract

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.

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Patrick Delorme, Vincent Secherre. 2006-12-19. An analogue of the Cartan decomposition for p-adic reductive symmetric spaces. https://arxiv.org/abs/math/0612545

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