This is meant to be a comment to Reimundo's answer, but as it runs longer than comments allow I am posting it as an answer.

In the general situation when you think of $G$ as a symplectic manifold, and $K$ a Lagrangian submanifold, it is often possible to make the local symplectomorphism guaranteed by Weinstein's Lagrangian Neighbourhood theorem quite explicit.

Suppose $G$ is complex reductive and $K$ an maximal compact subgroup. Consider the following two (left) actions of $K$ on $G$: $\mathcal{L}_k(g)=kg$ and $\mathcal{R}_k(g)=gk^{-1}$, for $k\in K$ and $g\in G$. Suppose $G$ has a symplectic form $\omega$, for which both actions, $\mathcal{L}$ and $\mathcal{R}$ are Hamiltonian, with moment maps $\mu_{\mathcal{L}}$ and $\mu_{\mathcal{R}}$. Since the actions commute the moment map for one is invariant for the other. Since the actions are free $K$ is Lagrangian and both moment-maps map onto open subsets in $\mathbb{k}^*$.

Consider now $G$ as a $K$-principal bundle by means of the action $\mathcal{L}$. This bundle is of course locally trivial, and it is not hard to see that one can in fact use $\mu_{\mathcal{R}}$ as the quotient map. Moreover this gives a symplectomorphism from $G$ to $K\times \mu_{\mathcal{R}}(G)\subset T^*K$. If $\mu_{\mathcal{R}}(G)$ contains $0$ this provides you with the local symplectomorphism guaranteed by the Lagrangian neighbourhood theorem; if $\mu_{\mathcal{R}}$ is surjective it gives a global symplectomorphism $G\cong T^*K$. This is the case for the Kähler structure provided by the polar decomposition and a choice of a metric, but also for at least a fair amount of the Kähler structures coming from affine embeddings.

anyembedding? $\endgroup$ – Reimundo Heluani Oct 19 '11 at 23:29