AN ANALOGUE OF THE CARTAN DECOMPOSITION FOR p-ADIC SYMMETRIC SPACES OF SPLIT p-ADIC REDUCTIVE GROUPS

作者：Delorme Patrick^*; Secherre Vincent

来源：Pacific Journal of Mathematics, 2011, 251(1): 1-21.

DOI：10.2140/pjm.2011.251.1

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

Let k be a nonarchimedean locally compact field of residue characteristic p, let G be a connected reductive group defined over k, let sigma be an involutive k-automorphism of G, and H an open k-subgroup of the fixed points group of sigma. 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 sigma - 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).

• 出版日期2011-5