# Cuspidal representation

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.

## Formulation

Let *G* be a reductive algebraic group over a number field *K* and let **A** denote the adeles of *K*. The group *G*(*K*) embeds diagonally in the group *G*(**A**) by sending *g* in *G*(*K*) to the tuple (*g*_{p})_{p} in *G*(**A**) with *g*=*g*_{p} for all (finite and infinite) primes *p*. Let *Z* denote the centre of *G* and let ω be a continuous unitary character from *Z*(*K*) \ Z(**A**)^{×} to **C**^{×}.
Fix a Haar measure on *G*(**A**) and let *L*^{2}_{0}(*G*(*K*) \ *G*(**A**), ω) denote the Hilbert space of measurable complex-valued functions, *f*, on *G*(**A**) satisfying

*f*(γ*g*) =*f*(*g*) for all γ ∈*G*(*K*)*f*(*gz*) =*f*(*g*)ω(*z*) for all*z*∈*Z*(**A**)- for all unipotent radicals,
*U*, of all proper parabolic subgroups of*G*(**A**).

The vector space *L*^{2}_{0}(*G*(*K*) \ *G*(**A**), ω) is called the **space of cusp forms with central character ω** on *G*(**A**). A function appearing in such a space is called a **cuspidal function**.

A cuspidal function generates a unitary representation of the group *G*(**A**) on the complex Hilbert space generated by the right translates of *f*. Here the action of *g* ∈ *G*(**A**) on is given by

- .

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces

where the sum is over irreducible subrepresentations of *L*^{2}_{0}(*G*(*K*) \ *G*(**A**), ω) and the *m*_{π} are positive integers (i.e. each irreducible subrepresentation occurs with *finite* multiplicity). A **cuspidal representation of G(A)** is such a subrepresentation (π,

*V*) for some

_{π}*ω*.

The groups for which the multiplicities *m*_{π} all equal one are said to have the multiplicity-one property.

## See also

## References

- James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty.
*Lectures on Automorphic L-functions*(2004), Chapter 5.