Based loop groups as homogeneous spaces

1. Nov 2, 2012

Kreizhn

1. The problem statement, all variables and given/known data
Let G be a compact connected Lie group define the loop group and the based loop group as $LG = \{ \gamma \in C^\infty(S^1,G) \}, \Omega G = \{ \gamma \in LG : \gamma(e_{S^1}) = e_G \}$ (choose whatever identification of the circle $S^1$ you like ). Show that $\Omega G$ is a homogeneous space over $LG$.

2. Relevant equations

The loop group LG is a group under pointwise multiplication, pointwise inverse, and has as its identity the constant loop at $e_G$.

3. The attempt at a solution
The idea is supposed to be as follows: LG acts on $\Omega G$ by conjugation. Then the stabilizer at identity of $\Omega G$ (also the constant loop at identity) is supposed to be the set of all constant loops, which we may identify with G. Then $\Omega G = LG/G$ which turns $\Omega G$ into a homogeneous space.

However, I think it's pretty clear that the stabilizer of any group action on a subgroup by conjugation is the entire group. Where do the constant functions come in?