In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If g denotes the Lie algebra of G , the corresponding action of G on g ∗ , the dual space to g , is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G .
The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G a basic role in their representation theory is played by coadjoint orbit. In the Kirillov method of orbits, representations of G are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G , which again may be complicated, while the orbits are relatively tractable.
Let G be a Lie group and g be its Lie algebra. Let A d : G → A u t ( g ) denote the adjoint representation of G . Then the coadjoint representation K : G → A u t ( g ∗ ) is defined as A d ∗ ( g ) := A d ( g − 1 ) ∗ . More explicitly,
⟨ K ( g ) F , Y ⟩ = ⟨ F , A d ( g − 1 ) Y ⟩ for
g ∈ G , Y ∈ g , F ∈ g ∗ , where ⟨ F , Y ⟩ denotes the value of a linear functional F on a vector Y .
Let K ∗ denote the representation of the Lie algebra g on g ∗ induced by the coadjoint representation of the Lie group G . Then for X ∈ g , K ∗ ( X ) = − a d ( X ) ∗ where a d is the adjoint representation of the Lie algebra g . One may make this observation from the infinitesimal version of the defining equation for K above, which is as follows :
⟨ K ∗ ( X ) F , Y ⟩ = ⟨ F , − a d ( X ) Y ⟩ for
X , Y ∈ g , F ∈ g ∗ . .
A coadjoint orbit Ω := O ( F ) for F in the dual space g ∗ of g may be defined either extrinsically, as the actual orbit K ( G ) ( F ) inside g ∗ , or intrinsically as the homogeneous space G / S t a b ( F ) where S t a b ( F ) is the stabilizer of F 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 g ∗ and carry a natural symplectic structure. On each orbit Ω , there is a closed non-degenerate G -invariant 2-form σ Ω inherited from g in the following manner. Let B F be an antisymmetric bilinear form on g defined by,
B F ( X , Y ) := ⟨ F , [ X , Y ] ⟩ , X , Y ∈ g Then one may define σ Ω ∈ H o m ( Λ 2 ( Ω ) , R ) by
σ Ω ( F ) ( K ∗ ( X ) ( F ) , K ∗ ( Y ) ( F ) ) := B F ( X , Y ) .
The well-definedness, non-degeneracy, and G -invariance of σ Ω follow from the following facts:
(i) The tangent space T F ( Ω ) may be identified with g / s t a b ( F ) , where s t a b ( F ) is the Lie algebra of S t a b ( F ) .
(ii) The kernel of B F is exactly s t a b ( F ) .
(iii) B F is invariant under S t a b ( F ) .
σ Ω is also closed. The canonical 2-form σ Ω is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit.
The coadjoint action on a coadjoint orbit ( Ω , σ Ω ) is a Hamiltonian G -action with moment map given by Ω ↪ g ∗ .
