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. 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 center of G and let ω be a continuous unitary character from Z(K) \ Z(A)&times; to C&times;. 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 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.
 * 1) f(γg) = f(g) for all γ ∈ G(K)
 * 2) f(gz) = f(g)ω(z) for all z ∈ Z(A)
 * 3) $$\int_{Z(\mathbf{A})G(K)\,\setminus\,G(\mathbf{A})}|f(g)|^2\,dg < \infty$$
 * 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).

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$\pi$ 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.