Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O\_q(Q\_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do for reals to find a Cartan decomposition?


Any reference would be highly helpful.



Indefinite orthogonal groups over p-adics
You should have a look at the following article by Delorme and Secherre :


Delorme, Patrick; Sécherre, Vincent, An analogue of the Cartan decomposition for $p$-adic symmetric spaces of split $p$-adic reductive groups. 
Pacific J. Math. 251 (2011), no. 1, 1–21. 


Aso see :


Y. Benoist and H. Oh, "Polar decomposition for $p$-adic symmetric spaces", Int. Math. Res. Not. 2007:24 (2007), article ID rnm121. 


