Let $G$ be a compact connected Lie group, and consider its cotangent bundle $T^*G$. There are two ways of viewing this space.
- Using left translation, we can trivialize $T^*G\cong G\times{\frak g}^*$. (Where ${\frak g}$ is the Lie algebra of $G$.) Moreover, $G$ acts on ${\frak g}^*$ via the coadjoint action, so we can consider the semi-direct product $$G\ltimes_{{\rm Ad}} {\frak g}^*.$$ This gives a Lie group structure on $T^*G$.
- On the other hand, there is a diffeomorphism $T^*G\cong G_{\Bbb C}$, where $G_{\Bbb C}$ is the complexification of $G$. (It comes from the map $G\times{\frak g}\to G_{\Bbb C},(g,X)\mapsto ge^{iX}$.) This gives a second Lie group structure on $T^*G$.
Does these two group structures have any interesting relation?
I don't have a counterexample, but I don't suspect the group structures to be the same. However can we still say something about their relation? In particular:
- Can we construct the group structure of $G_{\Bbb C}$ in a natural way from the one on $G\ltimes_{{\rm Ad}} {\frak g}^*$?
- Is there an interesting subset of $T^*G$ on which they coincide?