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.
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 (gp)p in G(A) with g=gp 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 L20(G(K) \ G(A), ω) denote the Hilbert space of measurable complex-valued functions, f, on G(A) satisfying
- f(γg) = 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 L20(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.
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 L20(G(K) \ G(A), ω) and the mπ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, Vπ) for some ω.
The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.
- James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Chapter 5.