# References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. However, I am not able to find some organic treatment of the differential geometry of the orbits of a smooth (analytic) action of an infinite-dimensional Banach-Lie group $\mathcal{G}$ on Banach manifolds $\mathcal{M}$. I am particularly interested in the case of non-proper actions, and I would like to know if and under what assumptions the orbits are Banach manifolds.

Are there articles/books developing the subject?

Thank You.

EDIT

Here and in "Bourbaki: Lie groups and Lie algebras, chapters 2 and 3", I found that, whenever the isotropy subgroup $\mathcal{G}_{m}$ at $m\in\mathcal{M}$ is a split Lie subgroup of $\mathcal{G}$ (that is, a subgroup which is a Lie group in the subspace topology), then $\mathcal{G}/\mathcal{G}_{m}$ is an analytic Banach manifold, and $\pi\colon\mathcal{G}\rightarrow\mathcal{G}_{m}$ is a submersion. Clearly, there is a bijection $\gamma$ from $\mathcal{G}/\mathcal{G}_{m}$ to the orbit $\mathcal{G}\cdot m$ through $m\in\mathcal{M}$. What do I have to do to ensure that the bijection $\gamma$ turns $\mathcal{G}\cdot m$ into a Banach manifold?

• I would also like a nice source for this. Many standard texts for Banach manifolds do not talk much about Banach Lie groups and their actions. – user98602 Apr 12 '16 at 21:14