Automorphic Form - Automorphic Representations

Automorphic Representations

The subsequent notion of automorphic representation has proved of great technical value for 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 approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 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 groups, with specific enveloping algebra representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.