An analogue of the Cartan decomposition forp-adic symmetric spaces of splitp-adic reductive groups

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 \Gk 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 .