Cuspidal representation

Summarize Timeline Top Qs Fact Check

Let ${\displaystyle G}$ be a reductive algebraic group over a number field ${\displaystyle K}$ and let ${\displaystyle \mathbb {A} }$ denote the adeles of ${\displaystyle K}$. The group ${\displaystyle G(K)}$ embeds diagonally in the group ${\displaystyle G(\mathbb {A} )}$ by sending ${\displaystyle g}$ in ${\displaystyle G(K)}$ to the tuple ${\displaystyle (g_{p})_{p}}$ in ${\displaystyle G(\mathbb {A} )}$ with ${\displaystyle g=g_{p}}$ for all (finite and infinite) primes ${\displaystyle p}$.

Let ${\displaystyle Z}$ denote the center of ${\displaystyle G}$ and let ${\displaystyle \omega }$ be a continuous unitary character from ${\displaystyle Z(K)\setminus Z(\mathbb {A} )^{\times }}$ to ${\displaystyle \mathbb {C} ^{\times }}$. Fix a Haar measure on ${\displaystyle G(\mathbb {A} )}$ and let ${\displaystyle L_{0}^{2}(G(K)\setminus G(\mathbb {A} ),\omega )}$ denote the Hilbert space of complex-valued measurable functions ${\displaystyle f}$ on ${\displaystyle G(\mathbb {A} )}$ satisfying

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

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

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

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

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

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

• Jacquet module