Cuspidal representation

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, cuspidal representations are certain <a href="/facts/Group_representation/FsuuDagQ">representations</a> of <a href="/facts/Algebraic_groups/DDtBCIvh">algebraic groups</a> that occur discretely in 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
 spaces. The term cuspidal is derived, at a certain distance, from the <a href="/facts/Cusp_form/m0zHUrmc">cusp forms</a> of classical <a href="/facts/Modular_form/PcnbYR1f">modular form</a> theory. In the contemporary formulation of <a href="/facts/Automorphic_representation/H0Pn7MyP">automorphic representations</a>, representations take the place of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a>; these representations may be of <a href="/facts/Adelic_algebraic_group/lh1ccLJB">adelic algebraic groups</a>.
When the group is the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> 
 
 
 
 
 GL
 
 2
 
 
 
 
 {\displaystyle \operatorname {GL} _{2}}
 
, the cuspidal representations are directly related to cusp forms and <a href="/facts/Maass_form/ep0I4xyk">Maass forms</a>. For the case of cusp forms, each <a href="/facts/Hecke_eigenform/kde8asT6">Hecke eigenform</a> (<a href="/facts/Newform/KgNk1KCW">newform</a>) corresponds to a cuspidal representation.

Cuspidal representation open-in-new

Cuspidal representation