Kähler structure on a complex reductive group Let $G$ be a complex reductive group, and $K$ a maximal compact subgroup (such that $K_{\mathbb{C}}=G$).  By the polar decomposition theorem one has that, as manifolds, $G\cong T^*K$.  The inherited symplectic structure is compatible with the complex structure, making $G$ into a Kähler manifold.
On the other hand $G$ is a smooth affine variety, and therefore inherits a Kähler structure from any embedding in an affine space.  The ring of regular functions of $G$ is described by the algebraic Peter-Weyl theorem, and affine embeddings are of course just given by choices of generators.
Can one obtain the Kähler structure coming from $T^*K$ by any of these affine embeddings?
 A: Isn't the answer no in the very simplest case? If $K$ is the circle group, then the Kähler structure on the cotangent bundle makes it metrically a cylinder $R \times S^{1}$. I believe this cylinder cannot be isometrically embedded in $C^n$ (apply the maximum modulus principal to the derivative of the map).
A: Let me give a simple argument until we think a better answer. Using Peter-Weyl you can choose an embedding of $G \subset \mathbb{A}^n$ such that $K$ is Lagrangian (using real representations for example), since $G = K_\mathbb{C} \simeq T^* K$ you obtain that $K$ is also Lagrangian in this manifold. Now you can use Weinstein's Lagrangian Neighborhood theorem (which gives a Lagrangian in a neighborhood of $K$, but perhaps this can be deformed by using that $K$ is maximal compact?).
A: 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.
