Cuspidal representation

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in $L^2$ spaces.Template:Sfnp 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 ${\mathrm{GL}}_2$, 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 $\omega$ be a continuous unitary character from $Z(K)\setminus Z(𝔸{)}^{\times }$ to ${ℂ}^{\times }$. Fix a Haar measure on $G(𝔸)$ and let $L_0^2(G(K)\setminus G(𝔸),\omega )$ denote the Hilbert space of complex-valued measurable functions $f$ on $G(𝔸)$ satisfying

1. $f(\gamma g)=f(g)$ for all $\gamma \in G(K)$,
2. $f(gz)=f(g)\omega (z)$ for all $z\in Z(𝔸)$,
3. ${\int }_{Z(𝔸)G(K)\setminus G(𝔸)}|f(g){|}^{2}dg<\mathrm{\infty }$,
4. ${\int }_{U(K)\setminus U(𝔸)}f(ug)du=0$ for all unipotent radicals $U$ of all proper parabolic subgroups of $G(𝔸)$ and $g\in G(𝔸)$.

The vector space $L_0^2(G(K)\setminus G(𝔸),\omega )$ is called the space of cusp forms with central character Template:Math 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 $g\in G(𝔸)$ on $V_f$ is given by

$(g\cdot u)(x)=u(xg),u(x)=\sum _j{c}_{j}f(x{g}_j)\in V_f.$

The space of cusp forms with central character $\omega$ decomposes into a direct sum of Hilbert spaces

$L_0^2(G(K)\setminus G(𝔸),\omega )={\widehat{⨁}}_{(\pi ,V_{\pi })}{m}_{\pi }{V}_{\pi }$

where the sum is over irreducible subrepresentations of $L_0^2(G(K)\setminus G(𝔸),\omega )$ and the $m_{\pi }$ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of Template:Math$𝔸$Template:Math is such a subrepresentation $(\pi ,V_{\pi })$ for some $\omega$.

The groups for which the multiplicities $m_{\pi }$ all equal $1$ are said to have the multiplicity-one property.

See also

• Jacquet module

References

• Template:Cite book