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 $\mathbb {A}$ denote the adeles of $K$. The group $G(K)$ embeds diagonally in the group $G(\mathbb {A} )$ by sending $g$ in $G(K)$ to the tuple $(g_{p})_{p}$ in $G(\mathbb {A} )$ 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 $Z(K)\setminus Z(\mathbb {A} )^{\times }$ to $\mathbb {C} ^{\times }$. Fix a Haar measure on $G(\mathbb {A} )$ and let $L_{0}^{2}(G(K)\setminus G(\mathbb {A} ),\omega )$ denote the Hilbert space of complex-valued measurable functions $f$ on $G(\mathbb {A} )$ satisfying

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

The vector space $L_{0}^{2}(G(K)\setminus G(\mathbb {A} ),\omega )$ is called the space of cusp forms with central character ω on $G(\mathbb {A} )$. A function appearing in such a space is called a cuspidal function.

A cuspidal function generates a unitary representation of the group $G(\mathbb {A} )$ on the complex Hilbert space $V_{f}$ generated by the right translates of $f$. Here the action of $g\in G(\mathbb {A} )$ 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 $\omega$ decomposes into a direct sum of Hilbert spaces

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

where the sum is over irreducible subrepresentations of $L_{0}^{2}(G(K)\setminus G(\mathbb {A} ),\omega )$ and the $m_{\pi }$ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G($\mathbb {A}$) 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

• Cogdell, James W.; Kim, Henry H.; Murty, M. Ram (2004). Lectures on Automorphic L-functions. Fields Institute Monographs. Vol. 20. American Mathematical Society. ISBN 978-0-82183516-6.