×

An analogue of the Cartan decomposition for \(p\)-adic symmetric spaces of split \(p\)-adic reductive groups. (English) Zbl 1220.22003

Let \(k\) be a non-Archimedean locally compact field of residue characteristic \(p\), \(G\) be a connected reductive group defined over \(k\), \(\sigma\) be an involutive \(k\)-automorphism of \(G\), and \(H\) be an open \(k\)-subgroup of the fixed points group of \(\sigma\). Let \(G_k\) and \(H_k\) be the groups of \(k\)-points of \(G\) and \(H\). The authors obtain an analogue of the Cartan decomposition for the reductive symmetric space \(H_k\setminus G_k\) in the case where \(G\) is \(k\)-split and \(p\) is odd. This generalizes the results by O. Offen [Pac. J. Math. 215, No. 1, 97-149 (2004; Zbl 1059.22019)] for \(GL_n\) and gives a refinement of the results by Y. Benoist and H. Oh [Int. Math. Res. Not. 2007, No. 24, Article ID rnm121, 20 p. (2007; Zbl 1137.22008)] who obtained a polar decomposition for any \(p\)-adic reductive symmetric space.

MSC:

22E35 Analysis on \(p\)-adic Lie groups
PDFBibTeX XMLCite
Full Text: DOI