Cuspidal representation

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title: "Cuspidal representation" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["representation-theory-of-algebraic-groups"] topic_path: "general/representation-theory-of-algebraic-groups" source: "https://en.wikipedia.org/wiki/Cuspidal_representation" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 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 \operatorname{GL}_2, 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 center 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 complex-valued measurable functions, f, on G(A) satisfying

  1. f(γg) = f(g) for all γ ∈ G(K)
  2. f(gz) = f(g)ω(z) for all z ∈ Z(A)
  3. \int_{Z(\mathbf{A})G(K),\setminus,G(\mathbf{A})}|f(g)|^2,dg
  4. \int_{U(K),\setminus,U(\mathbf{A})}f(ug),du=0 for all unipotent radicals, U, of all proper parabolic subgroups of G(A) and g ∈ 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.

A cuspidal function generates a unitary representation of the group G(A) on the complex Hilbert space V_f generated by the right translates of f. Here the action of g ∈ G(A) on V_f is given by :(g \cdot u)(x) = u(xg), \qquad u(x) = \sum_j c_j f(xg_j) \in V_f.

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces :L^2_0(G(K)\setminus G(\mathbf{A}),\omega)=\widehat{\bigoplus}{(\pi,V\pi)}m_\pi V_\pi where the sum is over irreducible subrepresentations of L20(G(K) \ G(A), ω) and the m are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (, V) for some ω.

The groups for which the multiplicities m all equal one are said to have the multiplicity-one property.

References

  • James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Section 5 of Lecture 2.

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