# Symplectic structures on the grassmannian model of the based loop group

$\newcommand{\Ad}{\operatorname{Ad}}$

In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian model of $[S]U(n)$ is well known. It is mentioned in Pressley-Segal's Loop Groups that there is a Grassmannian model for general centerfree compact groups. Here is a brief summary:

Let $G$ be a compact, connected, centerfree group with Lie algebra $\mathfrak g$ and complexification $G_\mathbb C$. Let $L^+G_{\mathbb C}$ be the algebraic maps $\mathbb C^\times \to \mathbb G_\mathbb C$ which restrict to holomorphic maps on the interior of the unit disk, and $H=L^2(S^1,\mathfrak g_\mathbb C)$, with Grassmannian $Gr(H)$. If $\gamma \in L^+G_\mathbb C$, define an action on $Gr(H)$ by $$\gamma \cdot W = \{ \Ad_{\gamma(z)}f(z): f \in W \}, \qquad \gamma \in L^+G, f \in H.$$ It is known that $\Omega G$ embeds into $Gr(H)$.

Now both spaces have natural symplectic structures, and in the case where $G=SU(n)$ it is also known that those structures are compatible with the embedding. I believe I also have a proof that shows the structures are compatible for general compact groups, but it requires an embedding into $SU(n)$.

Is there an intrinsic way (ie without embedding into $SU(n)$) to see that the symplectic structures are compatible with the embedding?