In number theory, cuspidal representations are certain representations of algebraic groups 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.

Formulation

be a reductive algebraic group over a number field denote the adeles.

denote the center be a continuous unitary character. Fix a Haar measure denote the Hilbert space of complex-valued measurable functions satisfying

1. ,

2. ,

3. ,

4. for all unipotent radicals of all proper parabolic subgroups.

The vector space is called the space of cusp forms with central character . A function appearing in such a space is called a cuspidal function.

A cuspidal function generates a unitary representation. Here the action is given by

decomposes into a direct sum of Hilbert spaces

where the sum is over irreducible subrepresentations are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of .

are said to have the multiplicity-one property.

See also

• Jacquet module

References