Coadjoint Orbit

A coadjoint orbit is the orbit of a Lie group acting on the dual of its Lie algebra via the coadjoint representation. For a Lie group G with Lie algebra g, the coadjoint action on the dual space g* is defined by ⟨Ad*_g(ξ), X⟩ = ⟨ξ, Ad_{g⁻¹}(X)⟩ for ξ ∈ g*, X ∈ g, and g ∈ G. These orbits are the geometric objects that parametrize the irreducible unitary representations of the group in the orbit method, and they carry a canonical symplectic structure that makes them the phase spaces of classical mechanical systems with symmetry.

The distinction between adjoint orbits (in g) and coadjoint orbits (in g*) is often merely technical: for semisimple Lie algebras, the Killing form provides a canonical identification between g and g*, and the adjoint and coadjoint orbits coincide. For more general groups — including the affine groups, Virasoro group, and groups of diffeomorphisms — the coadjoint orbit picture is the correct one, and the geometry of g* encodes information that is invisible in the adjoint picture.

Every coadjoint orbit is a homogeneous symplectic manifold, and the symplectic form is exact and G-invariant. The dimension of a coadjoint orbit is always even, and the orbit method proposes that each orbit corresponds to a unitary representation of the group, with the orbit's symplectic volume related to the representation's dimension (or formal degree, in the infinite-dimensional case). This correspondence is exact for nilpotent and solvable groups, and approximately correct for compact semisimple groups, where the Borel-Weil theorem provides the rigorous geometric quantization.

Coadjoint orbits reveal that the dual of a Lie algebra is not merely a vector space of linear functionals — it is the natural home of the group's classical mechanics. The fact that representation theory, symplectic geometry, and Hamiltonian mechanics all converge on the same geometric objects suggests that these are not separate subjects but different languages for describing a single structure. The coadjoint orbit is where these languages meet.