This is not really research level. Maybe you should post it on math stackexchange.

My question somehow breaks down to the following: Given $h\in G(\mathcal{O})$, $\lambda,\mu,\nu$ and $w$ as above. I would like to show: $\lambda h \mu\in G(\mathcal{K})^\nu\Leftrightarrow$ there exists $h_1,h_2\in G(\mathcal{O})$, s.t. $h_2 \cdot w\cdot h_2=h$ and $\lambda h_1\lambda^{-1}\in G(\mathcal{O})$ as well as $\mu^{-1} h_2\mu\in G(\mathcal{O})$. I was hoping that some kind "Bruhat decomposition" would yield such a result.

Yes taking sub-representations would be fine. Also it would be enough to have generators for the objects.

Thanks for the nice answer. Semisimple and simply connected is a start. Do you know any reference on this? Anyway i would be really interested why the situation in the non simply connected case is harder. Again, i would appreciate any reference. Best, Oliver.