Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the reduced space $$S=J^{-1}(\mu)/G_\mu$$ is exactly $G/G_\mu$ where $\mu\in \frak g^*$ and $G_\mu$ is the isotropy subgroup of $G$ at the point $\mu$

What we must choose for the space $M$ such that the reduced space $S$ is exactly $G^\mathbb C/(G_\mu)^\mathbb C$ where $G^\mathbb C$ is the complexification of the Lie group $G$ and we have $G^\mathbb C\cong G\times\frak {g}^*$ and so $$G^\mathbb C\cong {T^*G}?$$

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