Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in [math]\displaystyle{ L^2 }[/math] 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 [math]\displaystyle{ \operatorname{GL}_2 }[/math], 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. [math]\displaystyle{ \int_{Z(\mathbf{A})G(K)\,\setminus\,G(\mathbf{A})}|f(g)|^2\,dg \lt \infty }[/math]
4. [math]\displaystyle{ \int_{U(K)\,\setminus\,U(\mathbf{A})}f(ug)\,du=0 }[/math] 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 [math]\displaystyle{ V_f }[/math] generated by the right translates of f. Here the action of g ∈ G(A) on [math]\displaystyle{ V_f }[/math] is given by

[math]\displaystyle{ (g \cdot u)(x) = u(xg), \qquad u(x) = \sum_j c_j f(xg_j) \in V_f }[/math].

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

[math]\displaystyle{ L^2_0(G(K)\setminus G(\mathbf{A}),\omega)=\widehat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi }[/math]

where the sum is over irreducible subrepresentations of L²₀(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_π) for some ω.

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

See also

• Jacquet module

References

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