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

Patrick Delorme and Vincent Sécherre

Vol. 251 (2011), No. 1, 1–21

DOI: 10.2140/pjm.2011.251.1

Abstract

Let k be a nonarchimedean locally compact field of residue characteristic p, let G be a connected reductive group defined over k, let σ be an involutive k-automorphism of G, and H an open k-subgroup of the fixed points group of σ. We denote by G_k and H_k the groups of k-points of G and H. We obtain an analogue of the Cartan decomposition for the reductive symmetric space H_k∖G_k in the case where G is k-split and p is odd. More precisely, we obtain a decomposition of G_k as a union of (H_k,K)-double cosets, where K is the stabilizer of a special point in the Bruhat–Tits building of G over k. This decomposition is related to the H_k-conjugacy classes of maximal σ-antiinvariant k-split tori in G. In a more general context, Benoist and Oh obtained a polar decomposition for any p-adic reductive symmetric space. In the case where G is k-split and p is odd, our decomposition makes more precise that of Benoist and Oh, and generalizes results of Offen for GL_n.

Keywords

p-adic reductive group, building, Cartan decomposition, symmetric space

Mathematical Subject Classification 2000

Primary: 22E35

Milestones

Received: 18 January 2010

Accepted: 8 July 2010

Published: 22 April 2011