Coadjoint representation

Let G be a Lie group and \mathfrak{g} be its Lie algebra. Let \mathrm{Ad} : G \rightarrow \mathrm{Aut}(\mathfrak{g}) denote the adjoint representation of G. Then the coadjoint representation \mathrm{Ad}^*: G \rightarrow \mathrm{GL}(\mathfrak{g}^*) is defined by :\langle \mathrm{Ad}^*_g \, \mu, Y \rangle = \langle \mu, \mathrm{Ad}^{-1}_{g} Y \rangle = \langle \mu, \mathrm{Ad}_{g^{-1}} Y \rangle for g \in G, Y \in \mathfrak{g}, \mu \in \mathfrak{g}^*, where \langle \mu, Y \rangle denotes the value of the linear functional \mu on the vector Y. Let \mathrm{ad}^* denote the representation of the Lie algebra \mathfrak{g} on \mathfrak{g}^* induced by the coadjoint representation of the Lie group G. Then the infinitesimal version of the defining equation for \mathrm{Ad}^* reads: :\langle \mathrm{ad}^*_X \mu, Y \rangle = \langle \mu, - \mathrm{ad}_X Y \rangle = - \langle \mu, [X, Y] \rangle for X,Y \in \mathfrak{g}, \mu \in \mathfrak{g}^* where \mathrm{ad} is the adjoint representation of the Lie algebra \mathfrak{g}. ==Coadjoint orbit==

A coadjoint orbit \mathcal{O}_\mu for \mu in the dual space \mathfrak{g}^* of \mathfrak{g} may be defined either extrinsically, as the actual orbit \mathrm{Ad}^*_G \mu inside \mathfrak{g}^*, or intrinsically as the homogeneous space G/G_\mu where G_\mu is the stabilizer of \mu with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated. The coadjoint orbits are submanifolds of \mathfrak{g}^* and carry a natural symplectic structure. On each orbit \mathcal{O}_\mu, there is a closed non-degenerate G-invariant 2-form \omega \in \Omega^2(\mathcal{O}_\mu) inherited from \mathfrak{g} in the following manner: :\omega_\nu(\mathrm{ad}^*_X \nu, \mathrm{ad}^*_Y \nu) := \langle \nu, [X, Y] \rangle , \nu \in \mathcal{O}_\mu, X, Y \in \mathfrak{g}. The well-definedness, non-degeneracy, and G-invariance of \omega follow from the following facts: (i) The tangent space \mathrm{T}_\nu \mathcal{O}_\mu = \{ -\mathrm{ad}^*_X \nu : X \in \mathfrak{g}\} may be identified with \mathfrak{g}/\mathfrak{g}_\nu, where \mathfrak{g}_\nu is the Lie algebra of G_\nu. (ii) The kernel of the map X \mapsto \langle \nu, [X, \cdot] \rangle is exactly \mathfrak{g}_\nu. (iii) The bilinear form \langle \nu, [\cdot, \cdot] \rangle on \mathfrak{g} is invariant under G_\nu. \omega is also closed. The canonical 2-form \omega is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit. Properties of coadjoint orbits The coadjoint action on a coadjoint orbit (\mathcal{O}_\mu, \omega) is a Hamiltonian G-action with momentum map given by the inclusion \mathcal{O}_\mu \hookrightarrow \mathfrak{g}^*. ==Examples==