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?



Kähler structure on a complex reductive group
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).


