Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^\*$ (where $\mathfrak{t}$ is the Lie algebra of the maximal torus). Let $O\_{\lambda}$ be a generic [coadjoint orbit](http://ncatlab.org/nlab/show/coadjoint+orbit) and $O\_{\lambda'}$ be another coadjoint orbit ($\lambda'\in \mathfrak{g}^\*$). When are the two representations $\rho \colon G \to GL(\Gamma(O\_{\lambda},G\times\_{\mu} \mathbb{C}))$ (where $\Gamma(O\_{\lambda},G\times\_{\mu} \mathbb{C})$ is the set of holomorphic sections) and $\rho \colon G \to GL(\Gamma(O\_{\lambda'},G\times\_{\xi} \mathbb{C}))$ (where $\xi:O\_{\lambda'}\to S^1$ is a character) [equivalent](http://en.wikipedia.org/wiki/Group_representation)?



When representation of two different coadjoint orbits are equivalent?
These representations are the same if the coadjoint orbits are the same. Your notation is a bit confusing, but the basic fact is this: every coadjoint orbit intersects the positive Weyl chamber in a single point. If that point is an integral weight, then by Borel-Weil, the holomorphic sections of the line bundle you've written down are the irreducible representation with that highest weight. If the point isn't integral, then there simply is no such line bundle (since the weight don't integrate to the torus), which corresponds to the fact that no finite dimension representation has that weight.


