# Where is action free?

1. Jul 22, 2007

### 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