The example I am looking at is the universal cover of $SL_2(\mathbb{R})$ which is a semisimple Lie group with infinite center. Using Knapp's definition of reductive Lie groups, we need to find a compact subgroup $K\subseteq G$ and an involution $\theta$, a Killing form $B$ satisfying compatibility condition as below:
Then in the case of $G:=\widetilde{SL_2(\mathbb{R})}$ the mechanism for semisimple Lie groups can be fit in, that means we can still define compact real form on $\mathfrak{g}$ and use the original (almost) the original Cartan involution as the global Cartan involution. We have as a consequence the Iwasawa decomposition $KAN$ with $K\simeq \mathbb{R}^1$.
But if we use the above definition it seems the original structure is completely shattered. Since the $SO(2, \mathbb{R})$ is no longer the maximal compact subgroup of $G$. So my question is twofold:
- Is $SL_2(\mathbb{R})$ under such definition a reductive group?
- If yes, what kind of $K, B, \theta$ we should choose?
- Once we fixed the choice, does the corresponding Iwasawa decomposition look like the above one?
Any references regarding this context is equally welcome.