Kirillov character formula

<h2 id="character-formula-for-compact-lie-groups">Character formula for compact Lie groups</h2>
<p>Let 
  
    
      
        λ
        ∈
        
          
            
              t
            
          
          
            ∗
          
        
      
    
    {\displaystyle \lambda \in {\mathfrak {t}}^{*}}
  
 be the <a href="/facts/Highest_weight/7N8q5sw0">highest weight</a> of an <a href="/facts/Group_representation/FsuuDagQ">irreducible representation</a> 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
, where 
  
    
      
        
          
            
              t
            
          
          
            ∗
          
        
      
    
    {\displaystyle {\mathfrak {t}}^{*}}
  
 is the <a href="/facts/Dual_(mathematics)/8jGuyQIU">dual</a> of the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of the <a href="/facts/Maximal_torus/f82Qln74">maximal torus</a>, and let 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 be half the sum of the positive <a href="/facts/Root_system/TFaEdwMU">roots</a>.
</p><p>We denote by 
  
    
      
        
          
            
              O
            
          
          
            λ
            +
            ρ
          
        
      
    
    {\displaystyle {\mathcal {O}}_{\lambda +\rho }}
  
 the coadjoint orbit through 
  
    
      
        λ
        +
        ρ
        ∈
        
          
            
              t
            
          
          
            ∗
          
        
      
    
    {\displaystyle \lambda +\rho \in {\mathfrak {t}}^{*}}
  
 and by 
  
    
      
        
          μ
          
            λ
            +
            ρ
          
        
      
    
    {\displaystyle \mu _{\lambda +\rho }}
  
 the 
  
    
      
        G
      
    
    {\displaystyle G}
  
-invariant <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> on 
  
    
      
        
          
            
              O
            
          
          
            λ
            +
            ρ
          
        
      
    
    {\displaystyle {\mathcal {O}}_{\lambda +\rho }}
  
 with total mass 
  
    
      
        dim
        ⁡
        π
        =
        
          d
          
            λ
          
        
      
    
    {\displaystyle \dim \pi =d_{\lambda }}
  
, known as the <a href="/facts/Liouville_measure/JvNrJkC7">Liouville measure</a>.  If 
  
    
      
        
          χ
          
            λ
          
        
      
    
    {\displaystyle \chi _{\lambda }}
  
 is the character of the <a href="/facts/Group_representation/FsuuDagQ">representation</a>, the Kirillov's character formula for compact Lie groups is given by
</p>

j
          
            1
            
              /
            
            2
          
        
        (
        X
        )
        
          χ
          
            λ
          
        
        (
        exp
        ⁡
        X
        )
        =
        
          ∫
          
            
              
                
                  O
                
              
              
                λ
                +
                ρ
              
            
          
        
        
          e
          
            i
            β
            (
            X
            )
          
        
        d
        
          μ
          
            λ
            +
            ρ
          
        
        (
        β
        )
        ,
        
        ∀
        
        X
        ∈
        
          
            g
          
        
      
    
    {\displaystyle j^{1/2}(X)\chi _{\lambda }(\exp X)=\int _{{\mathcal {O}}_{\lambda +\rho }}e^{i\beta (X)}d\mu _{\lambda +\rho }(\beta ),\;\forall \;X\in {\mathfrak {g}}}
  
,
<p>where 
  
    
      
        j
        (
        X
        )
      
    
    {\displaystyle j(X)}
  
 is the <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian</a> of the exponential map.
</p>
<h2 id="example-su2">Example: SU(2)</h2>
<p>For the case of <a href="/facts/SU(2)/Qgy2orcu">SU(2)</a>, the <a href="/facts/Highest_weight/7N8q5sw0">highest weights</a> are the positive half integers, and 
  
    
      
        ρ
        =
        1
        
          /
        
        2
      
    
    {\displaystyle \rho =1/2}
  
.  The coadjoint orbits are the two-dimensional <a href="/facts/Spheres/jywYLNM2">spheres</a> of radius 
  
    
      
        λ
        +
        1
        
          /
        
        2
      
    
    {\displaystyle \lambda +1/2}
  
, centered at the origin in 3-dimensional space.
</p><p>By the theory of <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a>, it may be shown that
</p>

∫
          
            
              
                
                  O
                
              
              
                λ
                +
                1
                
                  /
                
                2
              
            
          
        
        
          e
          
            i
            β
            (
            X
            )
          
        
        d
        
          μ
          
            λ
            +
            1
            
              /
            
            2
          
        
        (
        β
        )
        =
        
          
            
              sin
              ⁡
              (
              (
              2
              λ
              +
              1
              )
              X
              )
            
            
              X
              
                /
              
              2
            
          
        
        ,
        
        ∀
        
        X
        ∈
        
          
            g
          
        
        ,
      
    
    {\displaystyle \int _{{\mathcal {O}}_{\lambda +1/2}}e^{i\beta (X)}d\mu _{\lambda +1/2}(\beta )={\frac {\sin((2\lambda +1)X)}{X/2}},\;\forall \;X\in {\mathfrak {g}},}

j
        (
        X
        )
        =
        
          
            
              sin
              ⁡
              X
              
                /
              
              2
            
            
              X
              
                /
              
              2
            
          
        
      
    
    {\displaystyle j(X)={\frac {\sin X/2}{X/2}}}

<p>thus yielding the characters of <i>SU</i>(2):
</p>

χ
          
            λ
          
        
        (
        exp
        ⁡
        X
        )
        =
        
          
            
              sin
              ⁡
              (
              (
              2
              λ
              +
              1
              )
              X
              )
            
            
              sin
              ⁡
              X
              
                /
              
              2
            
          
        
      
    
    {\displaystyle \chi _{\lambda }(\exp X)={\frac {\sin((2\lambda +1)X)}{\sin X/2}}}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Weyl_character_formula/nbbNv8oN">Weyl character formula</a></li>
<li><a href="/facts/Localization_formula_for_equivariant_cohomology/RI8Tlv1u">Localization formula for equivariant cohomology</a></li></ul>

<ul><li>Kirillov, A. A., <i>Lectures on the Orbit Method</i>, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, 64, AMS, Rhode Island, 2004.</li></ul>

Kirillov character formula open-in-new

Kirillov character formula