Cartan decomposition of loop group Let $G$ be a complex reductive group. Let $LG$ and $L^+ G$ denote the formal loop spaces given by maps from the punctured formal disk and the formal disk, respectively, to $G$. The quotient $LG/L^+ G$ is known as the affine grassmannian of $G$.
I am looking for a reference or a proof of the well known fact that the $L^+ G$ orbits of the affine grassmannian are in bijection with dominant coweights of $G$. In other words, the double coset space
$$
L^+ G \backslash LG / L^+ G = X_*(T),
$$
for some maximal torus $T$.
In some places it is stated that this is a consequence of the Cartan decomposition of the loop group $LG$, but I do not see exactly why that is true.
 A: Apparently, this is based on a result by Iwahori and Matsumoto (Corollary 2.17 of [IM]). A modern proof of this result can be found on [DHLH]. This result is one of the main elements of the proof of the Hilbert-Mumford criterion (see page 52 of [GIT]).
What the theorem of Iwahori and Matsumoto asserts is that any element $g\in LG$ can be written as $g=h_1 t^\mu h_2$, for $h_1, h_2 \in L^+G$, $\mu: \mathbb{G}_m\rightarrow G$ a 1-parameter subgroup and $t^\mu$ the image of $t\in \mathbb{C}((t))$ by the associated map $\mu(\mathbb{C}((t))): \mathbb{C}((t))^\times \rightarrow LG$.
Since maximal tori are conjugate, the orbit $L^+G t^\mu L^+G$ associated to any 1-PS can be labeled by a coweight $X_*(T)$. Moreover, we can further identify two orbits if the coweights are conjugate, so we can quotient by the Weyl group and get $X_*(T)/W=X_*(T)^+$.
A good reference for this topic, where I originally found about the paper of Iwahori-Matsumoto is section 4.5 of the book [BD].
I still do not know if there's an easier way of proving this (i.e. without using the theorem of Iwahori-Matsumoto) and do not get in what way this is a Cartan decomposition. Anyway, this solves my question.
References:
[AHLH] Alper, J., Halpern-Leistner, D., & Heinloth, J. (2019). Cartan-Iwahori-Matsumoto decompositions for reductive groups. arXiv preprint arXiv:1903.00128.
[BD] Beilinson, A.; Drinfeld, V. Quantization of Hitchin's integrable system and Hecke eigensheaves. https://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf
[IM] Iwahori, Nagayoshi; Matsumoto, Hideya. On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups. Publications Mathématiques de l'IHÉS, Tome 25 (1965) , pp. 5-48. http://www.numdam.org/item/PMIHES_1965__25__5_0/
[GIT] Mumford, D. ;  Fogarty, J. ;  Kirwan, F.  Geometric invariant theory.
Third edition.
