I'm currently reading a set of notes on Poisson manifolds and dynamics. The notes adopt the approach of reduction by group actions on the cotangent bundle of a Lie group, $G$.
I just have two quick questions about a particular pair of group actions on $T^* G$ and the reduction(s).
Suppose $G=HK$ is a product of Lie subgroups of a Lie group $G$. The notes say that $H$ (resp. $K$) acts on the symplectic manifold $T^*G$ on the left (resp. right), and then goes on to reduce by $H$ on the left, followed by a reduction by $K$ on the right.
Are these actions well-known in this setting? Are they typically just the cotangent lifts of left-translation (resp. right-translation) of $G$ by $H$ (resp. $K$)?
If so, is there a nice way of thinking about (and computing) these reductions?
Thanks for any help provided!