Timeline for G-invariant functions on manifold for G compact
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| Dec 18, 2015 at 15:43 | comment | added | Olorin | @Thomas Thank you very much for your counter-example. But if $M$ is a co-adjoint orbit, it wouldn't be a problem. My setting is, that we have $k$ functions $h_1, \dots, h_k \colon \Phi(M) \to \mathbb{R}$, that make the generic co-adjoint orbits Liouville integrable. If $G \in \{ SO(n), U(n)\}$ and the orbits in $M$ are now co-isotropic, then $M$ should also be Liouville integrable. So I'm assuming that $\dim M = 2n$ and we find $n-k$ functions as mentioned in my first question. And now I'm trying to find the reason, why we find such functions $f_1, \dots, f_{n-k}$. | |
| Dec 18, 2015 at 14:33 | comment | added | Thomas | This is not correct. If $M$ is a co-adjoint orbit, it is an homogenous space, and every invariant function is constant. It is however true if $s$ denotes the dimension of the polytope $\Phi(M)\cap C,$ if $C$ is a Weyl chamber. | |
| Dec 18, 2015 at 12:37 | history | asked | Olorin | CC BY-SA 3.0 |