# Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in ${\displaystyle L^{2}}$ 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 ${\displaystyle \operatorname {GL} _{2}}$, 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 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

1. fg) = f(g) for all γ ∈ G(K)
2. f(gz) = f(g)ω(z) for all zZ(A)
3. ${\displaystyle \int _{Z(\mathbf {A} )G(K)\,\setminus \,G(\mathbf {A} )}|f(g)|^{2}\,dg<\infty }$
4. ${\displaystyle \int _{U(K)\,\setminus \,U(\mathbf {A} )}f(ug)\,du=0}$ for all unipotent radicals, U, of all proper parabolic subgroups of 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 ${\displaystyle V_{f}}$ generated by the right translates of f. Here the action of gG(A) on ${\displaystyle V_{f}}$ is given by

${\displaystyle (g\cdot u)(x)=u(xg),\qquad u(x)=\sum _{j}c_{j}f(xg_{j})\in V_{f}}$.

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

${\displaystyle L_{0}^{2}(G(K)\setminus G(\mathbf {A} ),\omega )={\widehat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }}$

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