Coadjoint representation
In mathematics, the coadjoint representation [math]\displaystyle{ K }[/math] of a Lie group [math]\displaystyle{ G }[/math] is the dual of the adjoint representation. If [math]\displaystyle{ \mathfrak{g} }[/math] denotes the Lie algebra of [math]\displaystyle{ G }[/math], the corresponding action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ \mathfrak{g}^* }[/math], the dual space to [math]\displaystyle{ \mathfrak{g} }[/math], is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on [math]\displaystyle{ G }[/math]. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups [math]\displaystyle{ G }[/math] a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of [math]\displaystyle{ G }[/math] are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of [math]\displaystyle{ G }[/math], which again may be complicated, while the orbits are relatively tractable.
Formal definition
Let [math]\displaystyle{ G }[/math] be a Lie group and [math]\displaystyle{ \mathfrak{g} }[/math] be its Lie algebra. Let [math]\displaystyle{ \mathrm{Ad} : G \rightarrow \mathrm{Aut}(\mathfrak{g}) }[/math] denote the adjoint representation of [math]\displaystyle{ G }[/math]. Then the coadjoint representation [math]\displaystyle{ \mathrm{Ad}^*: G \rightarrow \mathrm{Aut}(\mathfrak{g}^*) }[/math] is defined by
- [math]\displaystyle{ \langle \mathrm{Ad}^*_g \, \mu, Y \rangle = \langle \mu, \mathrm{Ad}_{g^{-1}} Y \rangle }[/math] for [math]\displaystyle{ g \in G, Y \in \mathfrak{g}, \mu \in \mathfrak{g}^*, }[/math]
where [math]\displaystyle{ \langle \mu, Y \rangle }[/math] denotes the value of the linear functional [math]\displaystyle{ \mu }[/math] on the vector [math]\displaystyle{ Y }[/math].
Let [math]\displaystyle{ \mathrm{ad}^* }[/math] denote the representation of the Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] on [math]\displaystyle{ \mathfrak{g}^* }[/math] induced by the coadjoint representation of the Lie group [math]\displaystyle{ G }[/math]. Then the infinitesimal version of the defining equation for [math]\displaystyle{ \mathrm{Ad}^* }[/math] reads:
- [math]\displaystyle{ \langle \mathrm{ad}^*_X \mu, Y \rangle = \langle \mu, - \mathrm{ad}_X Y \rangle = - \langle \mu, [X, Y] \rangle }[/math] for [math]\displaystyle{ X,Y \in \mathfrak{g}, \mu \in \mathfrak{g}^* }[/math]
where [math]\displaystyle{ \mathrm{ad} }[/math] is the adjoint representation of the Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math].
Coadjoint orbit
A coadjoint orbit [math]\displaystyle{ \mathcal{O}_\mu }[/math] for [math]\displaystyle{ \mu }[/math] in the dual space [math]\displaystyle{ \mathfrak{g}^* }[/math] of [math]\displaystyle{ \mathfrak{g} }[/math] may be defined either extrinsically, as the actual orbit [math]\displaystyle{ \mathrm{Ad}^*_G \mu }[/math] inside [math]\displaystyle{ \mathfrak{g}^* }[/math], or intrinsically as the homogeneous space [math]\displaystyle{ G/G_\mu }[/math] where [math]\displaystyle{ G_\mu }[/math] is the stabilizer of [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ \mathfrak{g}^* }[/math] and carry a natural symplectic structure. On each orbit [math]\displaystyle{ \mathcal{O}_\mu }[/math], there is a closed non-degenerate [math]\displaystyle{ G }[/math]-invariant 2-form [math]\displaystyle{ \omega \in \Omega^2(\mathcal{O}_\mu) }[/math] inherited from [math]\displaystyle{ \mathfrak{g} }[/math] in the following manner:
- [math]\displaystyle{ \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} }[/math].
The well-definedness, non-degeneracy, and [math]\displaystyle{ G }[/math]-invariance of [math]\displaystyle{ \omega }[/math] follow from the following facts:
(i) The tangent space [math]\displaystyle{ \mathrm{T}_\nu \mathcal{O}_\mu = \{ -\mathrm{ad}^*_X \nu : X \in \mathfrak{g}\} }[/math] may be identified with [math]\displaystyle{ \mathfrak{g}/\mathfrak{g}_\nu }[/math], where [math]\displaystyle{ \mathfrak{g}_\nu }[/math] is the Lie algebra of [math]\displaystyle{ G_\nu }[/math].
(ii) The kernel of the map [math]\displaystyle{ X \mapsto \langle \nu, [X, \cdot] \rangle }[/math] is exactly [math]\displaystyle{ \mathfrak{g}_\nu }[/math].
(iii) The bilinear form [math]\displaystyle{ \langle \nu, [\cdot, \cdot] \rangle }[/math] on [math]\displaystyle{ \mathfrak{g} }[/math] is invariant under [math]\displaystyle{ G_\nu }[/math].
[math]\displaystyle{ \omega }[/math] is also closed. The canonical 2-form [math]\displaystyle{ \omega }[/math] 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 [math]\displaystyle{ (\mathcal{O}_\mu, \omega) }[/math] is a Hamiltonian [math]\displaystyle{ G }[/math]-action with momentum map given by the inclusion [math]\displaystyle{ \mathcal{O}_\mu \hookrightarrow \mathfrak{g}^* }[/math].
Examples
See also
- Borel–Bott–Weil theorem, for [math]\displaystyle{ G }[/math] a compact group
- Kirillov character formula
- Kirillov orbit theory
References
- Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, ISBN:0821835300, ISBN:978-0821835302
External links
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