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 $\operatorname_2$, 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 ω be a continuous unitary character from ''Z''(''K'') \ Z(A)^× to C^×.
Fix a Haar measure on ''G''(A) and let ''L''²₀(''G''(''K'') \ ''G''(A), ω) denote the Hilbert space of complex-valued measurable functions, ''f'', on ''G''(A) satisfying
#''f''(γ''g'') = ''f''(''g'') for all γ ∈ ''G''(''K'')
#''f''(''gz'') = ''f''(''g'')ω(''z'') for all ''z'' ∈ ''Z''(A)
#$\int_, f(g), ^2\,dg < \infty$
#$\int_f(ug)\,du=0$ for all unipotent radicals, ''U'', of all proper parabolic subgroups of ''G''(A) and g ∈ ''G''(A).
The vector space ''L''²₀(''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 $V_f$ 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
:$L^2_0(G(K)\setminus G(\mathbf),\omega)=\widehat_m_\pi V_\pi$
where the sum is over irreducible subrepresentations of ''L''²₀(''G''(''K'') \ ''G''(A), ω) and the ''m'' are positive integers (i.e. each irreducible subrepresentation occurs with ''finite'' multiplicity). A cuspidal representation of ''G''(''A'') is such a subrepresentation (, ''V'') for some ''ω''.

The groups for which the multiplicities ''m''_{{pi all equal one are said to have the multiplicity-one property.

See also

* Jacquet module

References

*James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. ''Lectures on Automorphic L-functions'' (2004), Section 5 of Lecture 2.

Representation theory of algebraic groups