Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in ${\displaystyle L^{2}}$ spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups.

When the group is the general linear group ${\displaystyle \operatorname {GL} _{2}}$, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.