Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and analytic geometry over non-Archimedean fields" Berkovich claims that we have the equality: $$G(K)=G(R)T(K)G(R).$$ This might be referred to as the Cartan decomposition of $G(K)$ and is known when $K$ is a discretely valued field. Berkovich cites Theorem 7.3.4 of Bruhat and Tits paper "Groupes réductifs sur un corps local : I. Données radicielles valuées" but to my understanding this paper only deals with the case where $K$ is discretely valued. Does the Cartan decomposition hold for an arbitrary valued field? Where can I find a reference for this?
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$\begingroup$ I’m not sure with the exact statement, but something should be true for general(non-discretely) valued field. See arxiv.org/abs/1110.1362 for example. $\endgroup$– coLaideronnetteCommented Jul 15 at 11:05
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