Cuspidal representation

<h2 id="formulation">Formulation</h2>
Let G be a <a href="/facts/Reductive_group/z1zyJvUh">reductive</a> algebraic group over a <a href="/facts/Number_field/e0K0TXSh">number field</a> K and let A denote the <a href="/facts/Adele_group/lh1ccLJB">adeles</a> of K. The group G(K) embeds diagonally in the group G(A) by sending g in G(K) to the tuple (gp)p in G(A) with g = gp for all (finite and infinite) primes p. Let Z denote the <a href="/facts/Center_of_a_group/gqAg6B8I">center</a> of G and let ω be a <a href="/facts/Continuous_(mathematics)/n8CGmpos">continuous</a> <a href="/facts/Character_(mathematics)/M46SyYUp">unitary character</a> from Z(K) \ Z(A)× to C×.
Fix a <a href="/facts/Haar_measure/2IcmTcuU">Haar measure</a> on G(A) and let L20(G(K) \ G(A), ω) denote the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> of <a href="/facts/Complex_number/2w7mqI7e">complex</a>-valued <a href="/facts/Measurable_function/NgIHnb1b">measurable functions</a>, f, on G(A) satisfying

<ol><li>f(γg) = f(g) for all γ ∈ G(K)</li>
<li>f(gz) = f(g)ω(z) for all z ∈ Z(A)</li>
<li>
 
 
 
 
 ∫
 
 Z
 (
 
 A
 
 )
 G
 (
 K
 )
 
 ∖
 
 G
 (
 
 A
 
 )
 
 
 
 |
 
 f
 (
 g
 )
 
 
 |
 
 
 2
 
 
 
 d
 g
 <
 ∞
 
 
 {\displaystyle \int _{Z(\mathbf {A} )G(K)\,\setminus \,G(\mathbf {A} )}|f(g)|^{2}\,dg<\infty }
 
</li>
<li>
 
 
 
 
 ∫
 
 U
 (
 K
 )
 
 ∖
 
 U
 (
 
 A
 
 )
 
 
 f
 (
 u
 g
 )
 
 d
 u
 =
 0
 
 
 {\displaystyle \int _{U(K)\,\setminus \,U(\mathbf {A} )}f(ug)\,du=0}
 
 for all <a href="/facts/Unipotent_radical/xnJlzBR9">unipotent radicals</a>, U, of all proper <a href="/facts/Borel_subgroup/F80yLMqA">parabolic subgroups</a> of G(A) and g ∈ G(A).</li></ol>
The <a href="/facts/Vector_space/M1pxMLx2">vector space</a> L20(G(K) \ G(A), ω) is called the space of cusp forms with central character ω on G(A). A function appearing in such a space is called a cuspidal function. 
A cuspidal function generates a <a href="/facts/Unitary_representation/W1XRatG8">unitary representation</a> of the group G(A) on the complex Hilbert space 
 
 
 
 
 V
 
 f
 
 
 
 
 {\displaystyle V_{f}}
 
 generated by the right translates of f. Here the <a href="/facts/Group_action/Vh9VtUUw">action</a> of g ∈ G(A) on 
 
 
 
 
 V
 
 f
 
 
 
 
 {\displaystyle V_{f}}
 
 is given by

(
 g
 ⋅
 u
 )
 (
 x
 )
 =
 u
 (
 x
 g
 )
 ,
 
 u
 (
 x
 )
 =
 
 ∑
 
 j
 
 
 
 c
 
 j
 
 
 f
 (
 x
 
 g
 
 j
 
 
 )
 ∈
 
 V
 
 f
 
 
 
 
 {\displaystyle (g\cdot u)(x)=u(xg),\qquad u(x)=\sum _{j}c_{j}f(xg_{j})\in V_{f}}
 
.
The space of cusp forms with central character ω decomposes into a <a href="/facts/Direct_sum_of_Hilbert_spaces/Y6PBJnjg">direct sum of Hilbert spaces</a>

L
          
            0
          
          
            2
          
        
        (
        G
        (
        K
        )
        ∖
        G
        (
        
          A
        
        )
        ,
        ω
        )
        =
        
          
            
              
                ⨁
                ^
              
            
          
          
            (
            π
            ,
            
              V
              
                π
              
            
            )
          
        
        
          m
          
            π
          
        
        
          V
          
            π
          
        
      
    
    {\displaystyle L_{0}^{2}(G(K)\setminus G(\mathbf {A} ),\omega )={\widehat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }}

where the sum is over <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible</a> <a href="/facts/Subrepresentation/MEJ9CMMD">subrepresentations</a> of L20(G(K) \ G(A), ω) and the mπ are positive <a href="/facts/Integer/PUwwrawx">integers</a> (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, Vπ) for some ω.
The groups for which the multiplicities mπ all equal one are said to have the <a href="/facts/Multiplicity-one_property/PnTdudlC">multiplicity-one property</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Jacquet_module/RTYe5dQW">Jacquet module</a></li></ul>

<ul><li>James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Section 5 of Lecture 2.</li></ul>

Cuspidal representation open-in-new

Cuspidal representation