Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra, and fix a Cartan involution $\theta$ of $\mathfrak{g}_0$, which gives a Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Since the automorphism group $G=\mathrm{Aut}\mathfrak{g}_0$ has finite center ($\textbf{Right?}$), there is an automorphism $\Theta$ of $G$ whose differential is $\theta$, and the subgroup $K$ of the fixed points of $\Theta$ on $G$ is a maximal compact subgroup of $G$. Moreover, $G=Ke^{\mathfrak{p}_0}$ as a polar decomposition of $G$.

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\sqrt{-1}\mathfrak{g}_0$ be the complexification of $\mathfrak{g}_0$, and denote by $G_\mathbb{C}$ the universal complexification of $G$. Set $\mathfrak{u}_0=\mathfrak{k}_0+\sqrt{-1}\mathfrak{p}_0$ which is a compact Lie algebra. Let $U$ be the subgroup of $G_\mathbb{C}$ corresponding to Lie algebra $\mathfrak{u}_0$.

$\textbf{QUESTION}$

1) Is $U$ a compact dual of $G$?

2) Does the polar decomposition $U=Ke^{\sqrt{-1}\mathfrak{p}_0}$ hold? In particular, does $K\subseteq G\cap U$ hold?

universalcomplexification of $G$? $\endgroup$ – Mikhail Borovoi Mar 29 '17 at 16:58universalcomplexification is usually just called the complexification, with no modifier (although your terminology is eminently logical). $\endgroup$ – LSpice Apr 4 '18 at 2:04