Constant term of an automorphic form

Definition

Let $F$ be a global field, let $G$ be a connected reductive $F$ -group, and let $P=MN$ be a parabolic subgroup with unipotent radical $N$ . For an automorphic form $\phi$ on $G(F)\backslash G(\mathbb A_F)$ , where $\mathbb A_F$ is the adele ring, its constant term along $P$ is

\phi_P(g)= \int_{N(F)\backslash N(\mathbb A_F)}\phi(ng)\,dn.

The quotient is compact for unipotent $N$ , so the integral is well-defined for the standard classes of automorphic forms after fixing Haar measure. The result transforms as an automorphic function on the Levi subgroup $M$ , with a normalization depending on whether the factor $\delta_P^{-1/2}$ is included.

Cuspidality

An automorphic form is cuspidal exactly when $\phi_P=0$ for every proper parabolic subgroup $P$ . This vanishing removes all contributions arriving from lower-rank Levi subgroups.

Eisenstein series and truncation

Constant terms of Eisenstein series are finite sums involving standard intertwining operators. Their asymptotics control poles, residual representations, and the continuous spectrum. Arthur truncation subtracts selected constant terms in the cuspidal directions to make trace-formula kernels integrable.

References

Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, 1976, Chapter II.
James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings 4, 2005, §§7 and 13. Clay.