2 Slightly rephrased the question.

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 a particular embeddingany of these affine embeddings?

1

# 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 a particular embedding?