# Structure of a group acting on a Hilbert space

Assume a group $$G$$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $$H$$ in such a way that the orbits are closed and the quotient $$H/G$$ is isometric to a finite dimensional Riemannian manifold.

Edit after YCor comment: Is there a natural way to equip $$G$$ with a topology so that it is an infinite dimensional Lie Group with a Lie Algebra?

• $G$ can be a large discrete group, say acting transitively. – YCor Dec 28 '18 at 21:47
• Hi YCor. Could you please describe an example with that property? I haven't been able to construct one. – Sergio Zamora Jan 14 '19 at 17:26
• Endow $G=H$ with the discrete topology, and let it act by translations on $H$. So the quotient is a point, and $G$ is a 0-dimensional Lie group. This is not an interesting example, probably additional hypotheses would be needed to discard it, and motivating examples could help then. – YCor Jan 14 '19 at 17:36
• Your addition doesn't help resolve YCor's comment, take $H = H_1 \oplus H_1$ and $G = H_1 \oplus H_1^\delta$, the first term $H_1$ with the usual topology and the second with the discrete topology, considered as an additive group. – mme Jan 16 '19 at 19:31
• Hello Mike. If one uses the group of all translations, it could be equipped with the pointwise convergence topology and then it becomes homeomorphic to the Hilbert space. – Sergio Zamora Jan 16 '19 at 19:40