Coadjoint representation

In mathematics, the coadjoint representation ${\displaystyle K}${\displaystyle K} of a Lie group ${\displaystyle G}${\displaystyle G} is the dual of the adjoint representation. If ${\displaystyle {\mathfrak {g}}}${\displaystyle {\mathfrak {g}}} denotes the Lie algebra of ${\displaystyle G}${\displaystyle G}, the corresponding action of ${\displaystyle G}${\displaystyle G} on ${\displaystyle {\mathfrak {g}}^{*}}${\displaystyle {\mathfrak {g}}^{*}}, the dual space to ${\displaystyle {\mathfrak {g}}}${\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 ${\displaystyle G}${\displaystyle G}.

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

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

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

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

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

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

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

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

${\displaystyle \omega _{\nu }(\mathrm {ad} _{X}^{*}\nu ,\mathrm {ad} _{Y}^{*}\nu ):=\langle \nu ,[X,Y]\rangle ,\nu \in {\mathcal {O}}_{\mu },X,Y\in {\mathfrak {g}}}${\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 ${\displaystyle G}${\displaystyle G}-invariance of ${\displaystyle \omega }${\displaystyle \omega } follow from the following facts:

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

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

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

${\displaystyle \omega }${\displaystyle \omega } is also closed. The canonical 2-form ${\displaystyle \omega }${\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 ${\displaystyle ({\mathcal {O}}_{\mu },\omega )}${\displaystyle ({\mathcal {O}}_{\mu },\omega )} is a Hamiltonian ${\displaystyle G}${\displaystyle G}-action with momentum map given by the inclusion ${\displaystyle {\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}}${\displaystyle {\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}}.

• Borel–Bott–Weil theorem, for ${\displaystyle G}${\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 .

• Representation theory of Lie groups
• Symplectic geometry

• Articles to be expanded from November 2014
• All articles to be expanded
• Articles with empty sections from November 2014
• All articles with empty sections