Is the T x S^1 action on C^infinity functions free?

[Firefly!]

Let G = SU(n), T = maximal torus of G
(so T is the group of diagonal matrices in SU(n) with elements in S^1)

M = C^infinity functions f: S^1 -> G s.t. f(0)=1

There is a T x S^1 action on M. T acts by conjugation & S^1 actions by "rotation"; (e^it.f)(s)=f(s+t)/f(t).


I'm trying to find where this action is free...

I thought I could find the stabilizers for the different 'types' of f & then consider the factor groups. I'm not sure this is an appropriate way to address this problem.

eg. f is fixed by T precisely when f(s) is in T
eg. f is fixed by S^1 if f is a homomorphism S^1-> G

I'm not sure if these are all of the stabilizers I need to consider. I'm also having problems describing the corresponding factor groups M/{f(s) in T} & M/{f in Hom(S^1,G)}


Any thoughts on any of this?

Thanks

 

[fresh_42]

I would prefer a more explicit way of what the various homomorphisms actually do and how the objects they operate on are described. This way you leave it to the reader to verify your various statements.
Here are some example calculations:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/