I am looking for a reference to the following statement "Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$ in $G$ is equal to the centralizer of $K$ in $G$ and they are equal to the center of $G$." I have a proof but I think this is a known result, I can´t find a reference.
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1$\begingroup$ Do you mean real reductive group? compact subgroups also make sense in the $p$-adic case and are quite different. Also, I expect dense to be in the Hausdorff topology? (although, well, it seems it plays no role, as it's clear that the question just boils down to describe the centralizer of $K$) $\endgroup$– YCorCommented Feb 23, 2016 at 17:14
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$\begingroup$ Complex reductive group $\endgroup$– user88059Commented Feb 23, 2016 at 17:39
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$\begingroup$ In the complex case, every maximal compact subgroup is Zariski-dense, and the result follows. $\endgroup$– YCorCommented Feb 23, 2016 at 18:49
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$\begingroup$ Why it follows? $\endgroup$– user88059Commented Feb 23, 2016 at 19:03
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$\begingroup$ The centralizer of $H$ in $G$ is equal to the centralizer of $K$ in $G$ (because $H$ is dense in $K$), which is in turn is equal to the centralizer of of $G$ in $G$ (because $K$ is Zariski-dense in $G$, as @YCor has mentioned), that is, to the center of $G$. In general the center of $G$ is not equal to the centralizer of $H$ in $K$ (take $G=\mathbb{C}^*$ and let $H=K$ be the maxiamal compact subgroup of $\mathbb{C}^*$). $\endgroup$– Mikhail BorovoiCommented Feb 23, 2016 at 20:00
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