Cuspidal representation

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. Embed G(K) diagonally in G(A), for example with $x\in GL_{1}(\mathbf {Q} )=\mathbf {Q} ^{*}$ and $(x)_{p},(x)_{\infty }$ the corresponding elements of $\mathbf {Q} _{p}^{*},\mathbf {R} ^{*}$ then $(x)_{\infty }\prod _{p}(x)_{p}\in GL_{1}(\mathbf {Q} )\cap GL_{1}(\mathbf {A_{Q}} )$ ). 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)
$\int _{Z(\mathbf {A} )G(K)\,\setminus \,G(\mathbf {A} )}|f(g)|^{2}\,dg<\infty$
$\int _{U(K)\,\setminus \,U(\mathbf {A} )}f(ug)\,du=0$ for all unipotent radicals, U, of all proper parabolic subgroups of G(A).

This is called the space of cusp forms with central character ω on G(A). A function occurring in such a space is called a cuspidal function.

Such a cuspidal function generates an unitary representation of the group G(A) on $V_{f}$ the complex Hilbert space generated by the right translates of f where 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_{0}^{2}(G(K)\setminus G(\mathbf {A} ),\omega )={\hat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }

where the sum is over irreducible subrepresentations of L²₀(G(K) \ G(A), ω) and 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.

References

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