Editing Automorphic form (section)

==Automorphic representations==
{{See also|Cuspidal representation}}
The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with ''G'' an [[algebraic group]], treated as an [[adelic algebraic group]]. It does not completely include the automorphic form idea introduced above, in that the [[Adele ring|adelic]] approach is a way of dealing with the whole family of [[congruence subgroup]]s at once. Inside an ''L''<sup>2</sup> space for a quotient of the adelic form of ''G'', an automorphic representation is a representation that is an infinite [[tensor product]] of representations of [[p-adic group]]s, with specific [[Universal enveloping algebra|enveloping algebra]] representations for the [[infinite prime]](s). One way to express the shift in emphasis is that the [[Hecke operator]]s are here in effect put on the same level as the Casimir operators; which is natural from the point of view of [[functional analysis]]{{Citation needed|reason=unveriviable and unsufficient citation about the source|date=May 2017}}, though not so obviously for the number theory. It is this concept that is basic to the formulation of the [[Langlands philosophy]].