Coadjoint Orbits in Lie Groups

• Coadjoint orbits are orbits of the coadjoint action on a Lie algebra’s dual space, forming homogeneous symplectic manifolds with a canonical Kirillov–Kostant–Souriau form.
• They underpin geometric quantization and establish a direct link between irreducible unitary representations and the symplectic geometry of classical and quantum systems.
• Recent advances utilize coadjoint orbits to construct manifestly covariant worldline actions, enabling geometric classification of particle types in Minkowski, de Sitter, and anti-de Sitter spaces.

Coadjoint orbits are the orbits of the coadjoint action of a Lie group $G$ on the dual space $\mathfrak{g}^*$ of its Lie algebra. Each coadjoint orbit $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$ (with $\mathrm{Ad}_g^*$ the coadjoint action) is a homogeneous symplectic manifold equipped with a canonical Kirillov–Kostant–Souriau (KKS) symplectic form. In mathematical physics and representation theory they provide a unified approach to the phase-space geometry of classical systems, underlie geometric quantization, and realize the orbit method linking irreducible unitary representations to symplectic geometry. Coadjoint orbits also serve as the geometric underpinning for manifestly covariant particle worldline actions for spacetime symmetry groups, permitting an explicit and geometric classification of particle types and their dynamics in Minkowski, de Sitter, and anti-de Sitter backgrounds.

1. General Construction and Symplectic Structure

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and dual $\mathfrak{g}^*$. The coadjoint action $\mathrm{Ad}_g^*$ is defined by

$$\langle \mathrm{Ad}^*_g\,\phi,\,X \rangle = \langle \phi,\, \mathrm{Ad}_{g^{-1}} X \rangle,$$

for $\phi \in \mathfrak{g}^*,\, X \in \mathfrak{g}$. The orbit $\mathfrak{g}^*$0 of a reference point $\mathfrak{g}^*$1 is identified with $\mathfrak{g}^*$2, where $\mathfrak{g}^*$3 is the stabilizer subgroup.

Every coadjoint orbit carries a canonical G-invariant symplectic form, the Kirillov–Kostant–Souriau (KKS) form: $\mathfrak{g}^*$4 where $\mathfrak{g}^*$5 and $\mathfrak{g}^*$6 denote the induced tangent vectors.

The coadjoint orbit action for a curve $\mathfrak{g}^*$7 is given by: $\mathfrak{g}^*$8 with gauge identifications by the stabilizer $\mathfrak{g}^*$9. Consistency under large $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$0 gauge transformations enforces quantization of the symplectic structure (prequantization) and leads to quantized invariants such as spin.

Alternatively, a coadjoint orbit can be realized as a symplectic reduction: embed $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$1 into a larger symplectic manifold (e.g., $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$2), introduce moment map constraints $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$3, and reduce by the stabilizer to recover the KKS form.

2. Symplectic Dual Pairs and Orbit Correspondences

A major structural feature is the existence of symplectic dual pairs: pairs of commuting Hamiltonian group actions $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$4 on a symplectic manifold $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$5, with associated moment maps $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$6. If the moment map fibers are group orbits of the dual action, there is a canonical correspondence between coadjoint orbits of $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$7 and $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$8 within the images of the moment maps: $$\mathcal{O}^G_\phi = \{ \mathrm{Ad}_g^* \phi\,|\, g \in G \}$$9

A concrete example is $\mathrm{Ad}_g^*$0, $\mathrm{Ad}_g^*$1, acting on $\mathrm{Ad}_g^*$2, where projection to either $\mathrm{Ad}_g^*$3 or $\mathrm{Ad}_g^*$4 defines dual orbit correspondences between G and $\mathrm{Ad}_g^*$5. Analogous dual pairs for orthogonal and symplectic groups provide the geometric backbone for many modern reductions in classical mechanics and field theory.

3. Explicit Orbit Classification for Spacetime Isometry Groups

Minkowski (Poincaré), de Sitter, and Anti-de Sitter Groups

The explicit classification of coadjoint orbits for spacetime symmetry groups governs the possible classical phase spaces for particle degrees of freedom in various background geometries (Basile et al., 2023). Invariants under the coadjoint action correspond to conserved quantities such as mass, spin, and continuous-spin parameters. Physical particle species map onto specific orbits according to their stabilizers (little groups):

Poincaré Group ($\mathrm{Ad}_g^*$6)

• Massive ($\mathrm{Ad}_g^*$7): Orbit type with stabilizer $\mathrm{Ad}_g^*$8.
• Massless ($\mathrm{Ad}_g^*$9): Stabilizer $G$0, W² = 0.
• Continuous-spin ($G$1): Null momenta and aligned Pauli–Lubanski vector.
• Tachyonic ($G$2): Stabilizer $G$3.
• Null spinless: $G$4, degenerate coadjoint orbits.

de Sitter (SO(1,d)) and Anti-de Sitter (SO(2,d-1))

• dS: Massless, massive, and partially massless orbits; absence of continuous-spin and tachyonic spinning orbits.
• AdS: Rich structure including massive, massless, tachyonic, continuous-spin, conformal (singleton) boundary orbits, plus "bitemporal" AdS orbits where mass and spin become interchanged variables.
• Spinning orbits in (A)dS exhibit shortening phenomena: for certain parameter values, orbits merge, corresponding to "partially massless" orbits in dS and "continuous-spin" or "mixed-spin" orbits in AdS.

Novel findings include the identification of new classes: partially massless spinning particles in dS, a vast array of boundary and bitemporal entities in AdS, and explicit worldline actions encoding these orbits (Basile et al., 2023).

4. Manifestly Covariant Worldline Actions from Coadjoint Orbits

For each coadjoint orbit $G$5 of a spacetime isometry group, a manifestly covariant worldline action can be constructed. These actions encode all physical constraints directly in terms of the symplectic geometry of the coadjoint orbit:

• Minkowski (Poincaré) Spinning Particle (with auxiliary spinors/vectors $G$6):

$G$7

where $G$8 are worldline gauge fields imposing the Casimir and constraint relations associated with the orbit invariants.

• (A)dS Spinning Particle:

$G$9

with ambient coordinates $\mathfrak{g}$0 and auxiliary spin variables $\mathfrak{g}$1.

Elimination of the momenta and auxiliary variables yields higher-derivative Lagrangians on the base space, reproducing known models (Nambu-Goto, Polyakov, etc.) in appropriate limits. The explicit inclusion of all Casimir constraints in the action encodes mass, spin, continuous-spin, and other labels within the worldline formalism.

Worldline actions arising from singular (nilpotent) orbits ("soft" limits) produce boundary or singleton dynamics, and bitemporal orbits in AdS correspond to new classes of models not previously accessible.

5. Inclusion Relations and Degenerations: Orbit Closure and Physical Limits

The fine structure of coadjoint orbits includes inclusion and degeneration relations:

• Nilpotent orbit closures (Hasse diagrams, signed Young tableau reductions): Larger orbits contain smaller nilpotent orbits in their closure. E.g., in SO(2,2), the conformal singleton orbit is at the intersection of two massless scalar orbits at the AdS and bitemporal boundaries.
• Soft limits: Sending mass or spin invariants to zero causes massive or massive-spinning orbits to degenerate to massless or boundary orbits, mirroring physical "soft photon" or "soft graviton" phenomena and the emergence of boundary degrees of freedom.
• Semisimple+nilpotent decompositions: Orbits of the form $\mathfrak{g}$2, with $\mathfrak{g}$3 in the semisimple part and $\mathfrak{g}$4 nilpotent, include as their closure all "smaller" nilpotent orbits $\mathfrak{g}$5 appearing in the closure of $\mathfrak{g}$6. This governs shortening and representation-theoretic transitions.

These structural relationships govern both the classical dynamics of relativistic particles and the classification and unitarity of the associated quantum representations.

6. Physical and Representation-Theoretic Implications

Coadjoint orbits classified as above correspond, upon quantization, to irreducible unitary representations of the underlying symmetry group, directly implementing an orbit-type version of the Wigner classification (flat space) and its generalizations to (A)dS (Basile et al., 2023). Notable assignments include:

• Massive orbits map to principal series representations.
• Massless and partially massless: Shortening phenomena lead to discrete, highest/lowest weight, and complementary series representations.
• Singleton/conformal: Boundary orbits correspond to singleton (minimal) representations.

Novel orbits, such as those in "bitemporal" AdS or continuous-spin orbits, imply the existence of new classes of classical and quantum particles. The full inclusion structure maps transitions between physical regimes—massive to massless, bulk to boundary—to explicit geometrical operations in the orbit hierarchy.

In summary, the coadjoint orbit method provides a unified and geometric approach to the complete classification, construction, and dynamical realization of all known and many new classical and quantum particle types in arbitrary constant-curvature backgrounds, with explicit worldline actions, symplectic structures, and representation-theoretic mappings (Basile et al., 2023).