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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.

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    $\begingroup$ Perhaps you mean SO$_q$ rather than O$_q$, but regardless, the uniqueness of maximal compact subgroups over $\mathbf{R}$ breaks down completely over non-archimedean local fields $k$ (every compact subgroup of $G(k)$ lies in a maximal one, and the number of conjugacy classes of maximal compact subgroups is finite but generally $> 1$). There is a good notion of Cartan decomposition for the group of points of a connected semisimple group over a non-archimedean local field (see the work of Bruhat--Tits), but it has nothing to do with notions like Cartan involution. See Tits' article in Corvallis. $\endgroup$
    – user76758
    Jan 31, 2014 at 6:14

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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.

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