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# coadjoint orbit

Let $G$ be a Lie group, and $\mathfrak{g}$ its Lie algebra. Then $G$ has a natural action on $\mathfrak{g}^{*}$ called the coadjoint action, since it is dual to the adjoint action of $G$ on $\mathfrak{g}$. The orbits of this action are submanifolds of $\mathfrak{g}^{*}$ which carry a natural symplectic structure, and are in a certain sense, the minimal symplectic manifolds on which $G$ acts. The orbit through a point $\lambda\in\mathfrak{g}^{*}$ is typically denoted $\mathcal{O}_{\lambda}$.

The tangent space $T_{\lambda}\mathcal{O}_{\lambda}$ is naturally idenified by the action with $\mathfrak{g}/\mathfrak{r}_{\lambda}$, where $\mathfrak{r}_{\lambda}$ is the Lie algebra of the stabilizer of $\lambda$. The symplectic form on $\mathcal{O}_{\lambda}$ is given by $\omega_{\lambda}(X,Y)=\lambda([X,Y])$. This is obviously anti-symmetric and non-degenerate since $\lambda([X,Y])=0$ for all $Y\in\mathfrak{g}$ if and only if $X\in\mathfrak{r}_{\lambda}$. This also shows that the form is well-defined.

There is a close association between coadoint orbits and the representation theory of $G$, with irreducible representations being realized as the space of sections of line bundles on coadjoint orbits. For example, if $\mathfrak{g}$ is compact, coadjoint orbits are partial flag manifolds, and this follows from the Borel-Bott-Weil theorem.

## Mathematics Subject Classification

53D05*no label found*

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