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 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 embeds diagonally in the group G by sending g in G to the tuple p in G with g=gp for all primes p. Let Z denote the centre of G and let ω be a continuous unitary character from Z \ Z× to C×.
Fix a Haar measure on G and let L20 \ G denote the Hilbert space of measurable complex-valued functions, f, on G satisfying
  1. f = f for all γ ∈ G
  2. f = fω for all z ∈ Z
  3. for all unipotent radicals, U, of all proper parabolic subgroups of G.
The vector space L20 \ G is called the space of cusp forms with central character ω on G. A function appearing in such a space is called a cuspidal function.
A cuspidal function generates a unitary representation of the group G on the complex Hilbert space generated by the right translates of f. Here the action of g ∈ G 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 L20 \ G and the m are positive integers. A cuspidal representation of G is such a subrepresentation for some ω.
The groups for which the multiplicities m all equal one are said to have the multiplicity-one property.