In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in 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 , 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.
In detail, let be a connected reductive algebraic group over an algebraic number field . Let be the adele of . Let be the center of . Let be a Hecke character. The character can be regarded as a character over
There is a representation of (called right regular representation) on the space
The representation is defined by
A function in
is a cuspidal function if for every parabolic subgroup of
Denote the space of cuspidal functions by
It is easily shown that the space is stable under the right regular representation.
It can be shown that the subspace of cuspidal functions can be decomposed discretely, i.e. it can be written as a direct sum of represntations of . Those irreducible subrepresentations of are called cuspidal representations.