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Coadjoint representation
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<h2 id="formal-definition">Formal definition</h2> <p>Let G {\displaystyle G} be a Lie group and g {\displaystyle {\mathfrak {g}}} be its Lie algebra. Let A d : G → A u t ( g ) {\displaystyle \mathrm {Ad} :G\rightarrow \mathrm {Aut} ({\mathfrak {g}})} denote the <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a> of G {\displaystyle G} . Then the coadjoint representation A d ∗ : G → G L ( g ∗ ) {\displaystyle \mathrm {Ad} ^{*}:G\rightarrow \mathrm {GL} ({\mathfrak {g}}^{*})} is defined by </p> ⟨ A d g ∗ μ , Y ⟩ = ⟨ μ , A d g − 1 Y ⟩ = ⟨ μ , A d g − 1 Y ⟩ {\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 g ∈ G , Y ∈ g , μ ∈ g ∗ , {\displaystyle g\in G,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*},} <p>where ⟨ μ , Y ⟩ {\displaystyle \langle \mu ,Y\rangle } denotes the value of the linear functional μ {\displaystyle \mu } on the vector Y {\displaystyle Y} . </p><p>Let a d ∗ {\displaystyle \mathrm {ad} ^{*}} denote the representation of the Lie algebra g {\displaystyle {\mathfrak {g}}} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} induced by the coadjoint representation of the Lie group G {\displaystyle G} . Then the infinitesimal version of the defining equation for A d ∗ {\displaystyle \mathrm {Ad} ^{*}} reads: </p> ⟨ a d X ∗ μ , Y ⟩ = ⟨ μ , − a d X Y ⟩ = − ⟨ μ , [ X , Y ] ⟩ {\displaystyle \langle \mathrm {ad} _{X}^{*}\mu ,Y\rangle =\langle \mu ,-\mathrm {ad} _{X}Y\rangle =-\langle \mu ,[X,Y]\rangle } for X , Y ∈ g , μ ∈ g ∗ {\displaystyle X,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*}} <p>where a d {\displaystyle \mathrm {ad} } is the <a href="/facts/Adjoint_representation_of_a_Lie_algebra/bWG4c2su">adjoint representation of the Lie algebra</a> g {\displaystyle {\mathfrak {g}}} . </p> <h2 id="coadjoint-orbit">Coadjoint orbit</h2> <p>A coadjoint orbit O μ {\displaystyle {\mathcal {O}}_{\mu }} for μ {\displaystyle \mu } in the dual space g ∗ {\displaystyle {\mathfrak {g}}^{*}} of g {\displaystyle {\mathfrak {g}}} may be defined either extrinsically, as the actual <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> A d G ∗ μ {\displaystyle \mathrm {Ad} _{G}^{*}\mu } inside g ∗ {\displaystyle {\mathfrak {g}}^{*}} , or intrinsically as the <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> G / G μ {\displaystyle G/G_{\mu }} where G μ {\displaystyle G_{\mu }} is the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">stabilizer</a> of μ {\displaystyle \mu } with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated. </p><p>The coadjoint orbits are submanifolds of g ∗ {\displaystyle {\mathfrak {g}}^{*}} and carry a natural symplectic structure. On each orbit O μ {\displaystyle {\mathcal {O}}_{\mu }} , there is a closed non-degenerate G {\displaystyle G} -invariant <a href="/facts/2-form/T9gyHtMv">2-form</a> ω ∈ Ω 2 ( O μ ) {\displaystyle \omega \in \Omega ^{2}({\mathcal {O}}_{\mu })} inherited from g {\displaystyle {\mathfrak {g}}} in the following manner: </p> ω ν ( a d X ∗ ν , a d Y ∗ ν ) := ⟨ ν , [ X , Y ] ⟩ , ν ∈ O μ , X , Y ∈ 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}}} . <p>The well-definedness, non-degeneracy, and G {\displaystyle G} -invariance of ω {\displaystyle \omega } follow from the following facts: </p><p>(i) The tangent space T ν O μ = { − a d X ∗ ν : X ∈ g } {\displaystyle \mathrm {T} _{\nu }{\mathcal {O}}_{\mu }=\{-\mathrm {ad} _{X}^{*}\nu :X\in {\mathfrak {g}}\}} may be identified with g / g ν {\displaystyle {\mathfrak {g}}/{\mathfrak {g}}_{\nu }} , where g ν {\displaystyle {\mathfrak {g}}_{\nu }} is the Lie algebra of G ν {\displaystyle G_{\nu }} . </p><p>(ii) The kernel of the map X ↦ ⟨ ν , [ X , ⋅ ] ⟩ {\displaystyle X\mapsto \langle \nu ,[X,\cdot ]\rangle } is exactly g ν {\displaystyle {\mathfrak {g}}_{\nu }} . </p><p>(iii) The bilinear form ⟨ ν , [ ⋅ , ⋅ ] ⟩ {\displaystyle \langle \nu ,[\cdot ,\cdot ]\rangle } on g {\displaystyle {\mathfrak {g}}} is invariant under G ν {\displaystyle G_{\nu }} . </p><p> ω {\displaystyle \omega } is also <a href="/facts/Closed_and_exact_differential_forms/uvEUbzUK">closed</a>. The canonical <a href="/facts/2-form/T9gyHtMv">2-form</a> ω {\displaystyle \omega } is sometimes referred to as the <i>Kirillov-Kostant-Souriau symplectic form</i> or <i>KKS form</i> on the coadjoint orbit. </p> <h3>Properties of coadjoint orbits</h3> <p>The coadjoint action on a coadjoint orbit ( O μ , ω ) {\displaystyle ({\mathcal {O}}_{\mu },\omega )} is a <a href="/facts/Hamiltonian_action/Rujfsdud">Hamiltonian G {\displaystyle G} -action</a> with <a href="/facts/Momentum_map/Rujfsdud">momentum map</a> given by the inclusion O μ ↪ g ∗ {\displaystyle {\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}} . </p> <h2 id="examples">Examples</h2> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Borel%E2%80%93Bott%E2%80%93Weil_theorem/Y4BXqu5I">Borel–Bott–Weil theorem</a>, for G {\displaystyle G} a compact group</li> <li><a href="/facts/Kirillov_character_formula/xXsrXuRG">Kirillov character formula</a></li> <li><a href="/facts/Kirillov_orbit_theory/8lR0osJN">Kirillov orbit theory</a></li></ul> <ul><li><a href="/facts/Kirillov,_A.A./0v7MQ7Im">Kirillov, A.A.</a>, <i>Lectures on the Orbit Method</i>, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, Vol. 64, American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0821835300, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0821835302</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/CoadjointOrbit">"Coadjoint orbit"</a>. <i><a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></i>.</li></ul>