The coadjoint representation, action and orbits

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$.

Let $G$ be a finite-dimensional real Lie group with Lie algebra $\mathfrak{g}=T_{e}G$ and dual ${\mathfrak{g}}^{\ast }=\mathcal{L}(\mathfrak{g},\mathbb{R})$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). For $g\in G$ the coadjoint map is the linear map

${\mathrm{Ad}}_g^{\ast }:{\mathfrak{g}}^{\ast }⟶{\mathfrak{g}}^{\ast },{\mathrm{Ad}}_g^{\ast }\alpha :=\alpha \circ {\mathrm{Ad}}_{g^{-1}},$

so that $⟨{\mathrm{Ad}}_g^{\ast }\alpha ,\xi ⟩=⟨\alpha ,{\mathrm{Ad}}_{g^{-1}}\xi ⟩$ for every $\xi \in \mathfrak{g}$. The coadjoint action of $G$ on ${\mathfrak{g}}^{\ast }$ is

$G\times {\mathfrak{g}}^{\ast }⟶{\mathfrak{g}}^{\ast },(g,\alpha )⟼g\cdot \alpha :={\mathrm{Ad}}_g^{\ast }\alpha \mathrm{.}$

The family ${\mathrm{Ad}}^{\ast }:G\to \mathrm{GL}({\mathfrak{g}}^{\ast })$, $g\mapsto {\mathrm{Ad}}_g^{\ast }$, is the coadjoint representation. The coadjoint orbit of $\alpha \in {\mathfrak{g}}^{\ast }$ and its coadjoint stabilizer are the orbit and stabilizer, in the sense of Orbits, stabilizers, and orbit maps of smooth actions, of this action:

$G\cdot \alpha =\{{\mathrm{Ad}}_g^{\ast }\alpha :g\in G\},G_{\alpha }=\{g\in G:{\mathrm{Ad}}_g^{\ast }\alpha =\alpha \}\mathrm{.}$

The maps ${\mathrm{Ad}}_g^{\ast }$ are invertible, with $({\mathrm{Ad}}_g^{\ast }{)}^{-1}={\mathrm{Ad}}_{g^{-1}}^{\ast }$: composing ${\mathrm{Ad}}_g^{\ast }{\mathrm{Ad}}_h^{\ast }$ gives $\alpha \mapsto \alpha \circ {\mathrm{Ad}}_{h^{-1}}\circ {\mathrm{Ad}}_{g^{-1}}$, and ${\mathrm{Ad}}_{h^{-1}}{\mathrm{Ad}}_{g^{-1}}={\mathrm{Ad}}_{h^{-1}{g}^{-1}}$ because $\mathrm{Ad}$ is a group homomorphism (Adjoint is a smooth Lie-group representation), which identifies the composite with ${\mathrm{Ad}}_{gh}^{\ast }$. Hence

${\mathrm{Ad}}_{gh}^{\ast }={\mathrm{Ad}}_g^{\ast }\circ {\mathrm{Ad}}_h^{\ast },{\mathrm{Ad}}_e^{\ast }={\mathrm{id}}_{{\mathfrak{g}}^{\ast }},$

so the coadjoint action is a left action; the inverse $g^{-1}$ in the definition is exactly what makes it left rather than right. The action is jointly smooth. Indeed, in a fixed basis of $\mathfrak{g}$ and its dual, the matrix of ${\mathrm{Ad}}_g^{\ast }$ is the transpose of the matrix of ${\mathrm{Ad}}_{g^{-1}}$; the matrix entries of $g\mapsto {\mathrm{Ad}}_{g^{-1}}$ are smooth because $\mathrm{Ad}$ is a smooth representation and inversion in $G$ is smooth (Lie group), and the coordinates of $(g,\alpha )\mapsto \alpha \circ {\mathrm{Ad}}_{g^{-1}}$ are these smooth matrix entries paired with the coordinates of $\alpha$ (Conjugation and the adjoint representation of a Lie group). Thus the coadjoint action is a smooth left action in the sense of Smooth left actions of Lie groups, and the orbit and stabilizer above are those of a smooth action.

Differentiating the curve $t\mapsto {\exp }_G(-t\xi )\cdot \alpha$ at $t=0$ gives the infinitesimal formula

${\xi }_{{\mathfrak{g}}^{\ast }}(\alpha )(\eta )={\frac{d}{dt}\mid }_0⟨{\exp }_G(-t\xi )\cdot \alpha ,\eta ⟩=\alpha ([\xi ,\eta ]),\eta \in \mathfrak{g},$

for the fundamental vector field ${\xi }_{{\mathfrak{g}}^{\ast }}(\alpha )=\frac{d}{dt}{\mathrm{\mid }}_0{\exp }_G(-t\xi )\cdot \alpha$ of Fundamental vector fields for a left action: the derivative of ${\mathrm{Ad}}_{{\exp }_G(t\xi )}=e^{t{\mathrm{ad}}_{\xi }}$ is ${\mathrm{ad}}_{\xi }$ (Adjoint exponential identity), and ${\mathrm{ad}}_{\xi }\eta =[\xi ,\eta ]$. Equivalently $(\xi \cdot \alpha )(\eta )=-\alpha ([\eta ,\xi ])$ if one writes the infinitesimal coadjoint action, but the formulation above is the one used in this library. A definition of the dual spaces, of the adjoint representation, of orbits and of the $\exp (-t\xi )$ convention, but of no further structure, is involved; the coadjoint action applies verbatim to disconnected $G$, to $\alpha =0$, whose orbit is the singleton $\{0\}$, and to abelian $G$, where it is trivial.

Here ${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice and is used only through the supplied fundamental-vector-field convention and the adjoint-exponential identity; no further choice is made in this definition.