# Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of $G$. If $T$ is a maximal torus of $G$, there is a well known $S^1 \times T$ action on $\Omega G$ given as follows: If $(s,t) \in S^1 \times T$ and $\gamma \in \Omega G$ then $$(s,t) \cdot \gamma(\theta) = t\gamma(\theta+s)\gamma(s)^{-1} t^{-1}.$$

If $G^{\mathbb C}$ is the complexified Lie group, define $L_\text{alg} G^{\mathbb C}$ as the collection of all free loops which are restrictions of algebraic maps from $\mathbb C^* \to G^{\mathbb C}$. We then define the algebraic based loops as $\Omega_{\text{alg}}G = L_{\text{alg}} G^{\mathbb C}\cap \Omega G$, which also inherits the $S^1 \times T$ action from $\Omega G$.

There are several papers which make use of the fact that $\Omega G$ is $S^1\times T$-equivariantly homotopy equivalent to $\Omega_\text{alg} G$, but I have yet to find any references or proof of this fact in the literature. Does anyone know a reference to this proof, or have an outline of how to prove it?

This paper and this paper show that $\Omega SU(2) \simeq_{SU(2)} \Omega_\text{alg} SU(2)$, and much of the proof can be adapted to get close to $\Omega G \simeq_{S^1 \times G} \Omega G$ but there are some difficulties I have yet to overcome. Nonetheless, it seems like the result should already exist somewhere.

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