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Let $G$ be a connected Lie group and let $K\subseteq G$ be a maximal compact subgroup. If $H\subseteq G$ is any compact subgroup I want to show that there exists $g\in G$ such that $g^{-1}Hg\subseteq K$.

Proof Idea: Note that $H$ acts on the coset space $G/K$ by left multiplication. We will be done if we can find a global fixed point $gK$ such that $hgK=gK$ (and hence $g^{-1}hg\in K$) for all $h\in H$.

My Question: What general properties of this action guarantee that it has a fixed point? I see hints that this might be related to the Bruhat-Tits fixed point theorem.

Drew Armstrong
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  • maybe useful https://en.wikipedia.org/wiki/Maximal_compact_subgroup#Existence_and_uniqueness – Tim kinsella Feb 16 '18 at 07:46
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    The standard proof (at least, the standard differential-geometric proof) is to argue that in case $G$ is reductive, the quotient $X=G/K$ is simply-connected and has a $G$-invariant Riemannian metric of nonpositive curvature. From that, Cartan's fixed point theorem (which is a precursor of the one due to Bruhat and Tits) implies that every compact group $C$ of isometries of $X$ has a fixed point in $X$. Then use the Levi-Malcev theorem to deal with general connected Lie groups. The wikipedia link above gives you references and more detail. – Moishe Kohan Feb 16 '18 at 14:19

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