Coadjoint representation

In mathematics, the coadjoint representation ${\displaystyle K}$ of a Lie group ${\displaystyle G}$ is the dual of the adjoint representation. If ${\displaystyle {\mathfrak {g}}}$ denotes the Lie algebra of ${\displaystyle G}$, the corresponding action of ${\displaystyle G}$ on ${\displaystyle {\mathfrak {g}}^{*}}$, the dual space to ${\displaystyle {\mathfrak {g}}}$, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on ${\displaystyle G}$.

The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups ${\displaystyle G}$ a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of ${\displaystyle G}$ are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of ${\displaystyle G}$, which again may be complicated, while the orbits are relatively tractable.

Let ${\displaystyle G}$ be a Lie group and ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra. Let ${\displaystyle \mathrm {Ad} :G\rightarrow \mathrm {Aut} ({\mathfrak {g}})}$ denote the adjoint representation of ${\displaystyle G}$. Then the coadjoint representation ${\displaystyle \mathrm {Ad} ^{*}:G\rightarrow \mathrm {GL} ({\mathfrak {g}}^{*})}$ is defined by

${\displaystyle \langle \mathrm {Ad} _{g}^{*}\,\mu ,Y\rangle =\langle \mu ,\mathrm {Ad} _{g}^{-1}Y\rangle =\langle \mu ,\mathrm {Ad} _{g^{-1}}Y\rangle }$ for ${\displaystyle g\in G,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*},}$

where ${\displaystyle \langle \mu ,Y\rangle }$ denotes the value of the linear functional ${\displaystyle \mu }$ on the vector ${\displaystyle Y}$.

Let ${\displaystyle \mathrm {ad} ^{*}}$ denote the representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle {\mathfrak {g}}^{*}}$ induced by the coadjoint representation of the Lie group ${\displaystyle G}$. Then the infinitesimal version of the defining equation for ${\displaystyle \mathrm {Ad} ^{*}}$ reads:

${\displaystyle \langle \mathrm {ad} _{X}^{*}\mu ,Y\rangle =\langle \mu ,-\mathrm {ad} _{X}Y\rangle =-\langle \mu ,[X,Y]\rangle }$ for ${\displaystyle X,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*}}$

where ${\displaystyle \mathrm {ad} }$ is the adjoint representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$.

A coadjoint orbit ${\displaystyle {\mathcal {O}}_{\mu }}$ for ${\displaystyle \mu }$ in the dual space ${\displaystyle {\mathfrak {g}}^{*}}$ of ${\displaystyle {\mathfrak {g}}}$ may be defined either extrinsically, as the actual orbit ${\displaystyle \mathrm {Ad} _{G}^{*}\mu }$ inside ${\displaystyle {\mathfrak {g}}^{*}}$, or intrinsically as the homogeneous space ${\displaystyle G/G_{\mu }}$ where ${\displaystyle G_{\mu }}$ is the stabilizer of ${\displaystyle \mu }$ 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 ${\displaystyle {\mathfrak {g}}^{*}}$ and carry a natural symplectic structure. On each orbit ${\displaystyle {\mathcal {O}}_{\mu }}$, there is a closed non-degenerate ${\displaystyle G}$-invariant 2-form ${\displaystyle \omega \in \Omega ^{2}({\mathcal {O}}_{\mu })}$ inherited from ${\displaystyle {\mathfrak {g}}}$ in the following manner:

${\displaystyle \omega _{\nu }(\mathrm {ad} _{X}^{*}\nu ,\mathrm {ad} _{Y}^{*}\nu ):=\langle \nu ,[X,Y]\rangle ,\nu \in {\mathcal {O}}_{\mu },X,Y\in {\mathfrak {g}}}$.

The well-definedness, non-degeneracy, and ${\displaystyle G}$-invariance of ${\displaystyle \omega }$ follow from the following facts:

(i) The tangent space ${\displaystyle \mathrm {T} _{\nu }{\mathcal {O}}_{\mu }=\{-\mathrm {ad} _{X}^{*}\nu :X\in {\mathfrak {g}}\}}$ may be identified with ${\displaystyle {\mathfrak {g}}/{\mathfrak {g}}_{\nu }}$, where ${\displaystyle {\mathfrak {g}}_{\nu }}$ is the Lie algebra of ${\displaystyle G_{\nu }}$.

(ii) The kernel of the map ${\displaystyle X\mapsto \langle \nu ,[X,\cdot ]\rangle }$ is exactly ${\displaystyle {\mathfrak {g}}_{\nu }}$.

(iii) The bilinear form ${\displaystyle \langle \nu ,[\cdot ,\cdot ]\rangle }$ on ${\displaystyle {\mathfrak {g}}}$ is invariant under ${\displaystyle G_{\nu }}$.

${\displaystyle \omega }$ is also closed. The canonical 2-form ${\displaystyle \omega }$ is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.

The coadjoint action on a coadjoint orbit ${\displaystyle ({\mathcal {O}}_{\mu },\omega )}$ is a Hamiltonian ${\displaystyle G}$-action with momentum map given by the inclusion ${\displaystyle {\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}}$.

• Borel–Bott–Weil theorem, for ${\displaystyle G}$ a compact group
• Kirillov character formula
• Kirillov orbit theory

• Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, ISBN 0821835300, ISBN 978-0821835302

• "Coadjoint orbit". PlanetMath.