Coadjoint representation

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the coadjoint representation 
  
    
      
        K
      
    
    {\displaystyle K}
  
 of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is the <a href="/facts/Dual_representation/HJWTCk5d">dual</a> of the <a href="/facts/Adjoint_representation_of_a_Lie_group/bWG4c2su">adjoint representation</a>. If 
  
    
      
        
          
            g
          
        
      
    
    {\displaystyle {\mathfrak {g}}}
  
 denotes the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of 
  
    
      
        G
      
    
    {\displaystyle G}
  
, the corresponding action of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 on 
  
    
      
        
          
            
              g
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathfrak {g}}^{*}}
  
, the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> to 
  
    
      
        
          
            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 <a href="/facts/1-form/NQLbWmtD">1-forms</a> on 
  
    
      
        G
      
    
    {\displaystyle G}
  
.
</p><p>The importance of the coadjoint representation was emphasised by work of <a href="/facts/Alexandre_Kirillov/0v7MQ7Im">Alexandre Kirillov</a>, who showed that for <a href="/facts/Nilpotent_Lie_group/BJuochky">nilpotent Lie groups</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 a basic role in their <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> is played by coadjoint orbits.
In the Kirillov method of orbits, representations of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the <a href="/facts/Conjugacy_class/AFuPIFNW">conjugacy classes</a> of 
  
    
      
        G
      
    
    {\displaystyle G}
  
, which again may be complicated, while the orbits are relatively tractable.
</p>

Coadjoint representation open-in-new

Coadjoint representation