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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.

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    $\begingroup$ The Cartan decomposition states that $LG = L^+G T(K) L^+G$, and the result follows, as $T(K) = X_*(T)T(O)$. ($K$ denotes formal Laurent series, and $O$ formal power series.) I don't know a reference off the top of my head, but I would look in Xinwen Zhu's lectures on the geometry of the affine Grassmannian. $\endgroup$
    – Geordie Williamson
    Commented Jul 5, 2021 at 10:28
  • $\begingroup$ Thank you. I already read Zhu’s notes. What I’d like to know is precisely why the Cartan decomposition states that. $\endgroup$
    – G. Gallego
    Commented Jul 5, 2021 at 10:30

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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.

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  • $\begingroup$ The Cartan decomposition is a decomposition of a group (here $LG$) as a product $KAK$, where $K$ is a maximal compact and $A$ is a "maximal torus". $L^+G$ is a maximal compact of $LG$, hence the relation. That being said, I personally find the fact you stated easier to think about in the context of Bruhat decomposition, which gives another proof. $\endgroup$
    – dhy
    Commented Sep 15, 2021 at 18:24
  • $\begingroup$ Thanks for your answer! I also thought about that. What I don't see clearly is why $L^+ G$ is a maximal compact. I guess that it's fixed points of the involution induced by $t \mapsto t^{-1}$. I also do not see why you get LT as the maximal torus. Anyway, my question now is why then Iwahori-Matsumoto is not "trivial"? Could it be that the Cartan decomp. does not exist over any field? $\endgroup$
    – G. Gallego
    Commented Sep 16, 2021 at 20:39
  • $\begingroup$ It may be helpful to note that the original setting in which all of these theorems arose was for p-adic groups. If you take $G$ a split group over $\mathbb{Q}_p$, then the maximal compact subgroup of $G(\mathbb{Q}_p)$ is $G(\mathbb{Z}_p)$. For loop groups, $G(\mathbb{Q}_p)$ is replaced by $LG$ and $G(\mathbb{Z}_p)$ is replaced by $L^+G$. The resulting decomposition is called "Cartan decomposition" by analogy to the case of real Lie groups, but I don't know of any uniform proof in all settings. $\endgroup$
    – dhy
    Commented Sep 16, 2021 at 21:46

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