# Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$.

1. Let $\mathcal H^{\mathfrak g} = L^2(S^1,\mathfrak g)$ equipped with a polarization $\mathcal H^{\mathfrak g} = \mathcal H_+ \oplus \mathcal H_-$. Consider the corresponding Grassmannian $\text{Gr}^{\mathfrak g}$ consisting of linear subspaces $W \subseteq \mathcal H^{\mathfrak g}$ such that
1. Projection onto $\mathcal H_+$ is Fredholm and projection onto $\mathcal H_-$ is Hilbert-Schmidt.
2. $zW \subseteq W$ (where $z$ is the $S^1$ coordinate, thinking of elements of $\mathcal H^{\mathfrak g}$ in their Fourier basis $\{ z^k \}$. )

Then $\text{Gr}^{\mathfrak g}$ has a Kahler structure whose Kahler form is induced by $$\omega(X,Y) = -i \text{Trace}(X^*Y-Y^*X)$$ where $\omega$ is defined at $T_{\mathcal H_+} \text{Gr}^{\mathfrak g}$ as identified with the collection of Hilbert-Schmidt operators from $\mathcal H_+ \to \mathcal H_-$.

1. Let $\Omega G$ be the collection of identity preserving smooth maps $S^1 \to G$. Give this the Kahler form by specifying that at $\Omega \mathfrak g$ we have $$\omega(\gamma,\eta) = \frac1{2\pi} \int_{S^1} \langle \eta,\gamma' \rangle$$ where $\langle,\rangle$ is some $\text{Ad}$-invariant inner product on $\mathfrak g$.

It is known that $\Omega G$ is biholomorphic to $\text{Gr}^{\mathfrak g}$ under the map $\gamma \mapsto \gamma \mathcal H_+$ where $$\gamma \mathcal H_+ = \{ \text{Ad}_{\gamma(z)} X(z): X \in \mathcal H_+\}.$$ I wonder if the Kahler structures are similarly equivalent. I have tried to pullback the Kahler form under the above biholomorphism, but to no avail.

• Footnote 19 of arxiv.org/pdf/1503.06254v1.pdf is relevant to this question. It is exactly the failure of two Kahler structures being equivalent which constitutes the gap in Posthuma's attempt at constructing the WZW Segal CFT. – André Henriques Oct 15 '15 at 11:12