Cuspidal automorphic representation

Let $G$ be a connected reductive group over a global field $F$ . An automorphic form $f$ is cuspidal if

\int_{N(F)\backslash N(\mathbb A_F)} f(ng)\,dn=0

for every proper $F$ -parabolic subgroup $P=MN$ , with Levi subgroup $M$ and unipotent radical $N$ , and every $g\in G(\mathbb A_F)$ . A cuspidal automorphic representation is an irreducible representation occurring in the space generated by cuspidal automorphic forms.

Analytic position

After fixing a unitary central character or quotienting by the split center, cuspidal forms are square-integrable. Their representation space decomposes discretely with finite multiplicities:

L^2_{\mathrm{cusp}} \cong \widehat{\bigoplus}_{\pi} m_{\mathrm{cusp}}(\pi)\,\pi.

Thus the cuspidal spectrum lies in the discrete automorphic spectrum. The converse fails: the discrete spectrum can also contain residual representations.

Why every proper parabolic matters

The integral is the constant term along $P$ . Vanishing only along a chosen Borel is sufficient in some standard situations but is not the invariant general definition. The condition must be imposed only for parabolic subgroups defined over $F$ .

For an $F$ -anisotropic group modulo center there are no proper $F$ -parabolics, so every automorphic form is cuspidal by this definition.

Local versus global cuspidality

Cuspidality here is a global condition on constant terms. It is not the same as supercuspidality of a local component. A cuspidal automorphic representation may have principal series components at many places; indeed it is unramified at almost all finite places.

Langlands-program role

Cuspidal representations are the basic discrete global inputs. Eisenstein series built from cuspidal data on Levi subgroups organize the remainder of the automorphic spectrum, while global reciprocity seeks arithmetic parameters for the cuspidal pieces.

References

Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, 1976. DOI.
A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” Proc. Sympos. Pure Math. 33, part 1, 1979.