I'm reading a paper on complex semi-simple algebraic group geometry at the moment, but finding the going a bit tough since I'm missing alot of the prerequisites. Firstly, the author refers to a *principal embedding* of one Lie group into another. I guess that this is an embedding of one group into another that is maximal in some sense. The paper states that in the dual Lie algebra setting this means that ${\frak g}$ is obtained from ${\frak g_0}$ by adding a node to the Dynkin diagram. The family of examples used is that of the embedding of $U_{N-1}$ into $SU_{N}$. I've searched the web but can't find a principal embedding defn, could someone point me in the right direction.

Secondly, if $G_0$ is principal embedded into $G$, is there some result about the quotient $G/G_0$ having a complex structure? This works for example $CP^{N-1} = SU_N/U_{N-1}$.