Eisenstein series on a reductive group

Let $G$ be a connected reductive group over a global field $F$ , let $P=MN$ be a parabolic subgroup with Levi subgroup $M$ and unipotent radical $N$ , and let $\phi_\lambda$ be a suitable $K$ -finite section of a normalized representation induced from cuspidal automorphic data on $M(\mathbb A_F)$ . The associated Eisenstein series is

E(g,\phi,\lambda) = \sum_{\gamma\in P(F)\backslash G(F)} \phi_\lambda(\gamma g),

initially for $\operatorname{Re}(\lambda)$ in a sufficiently positive chamber.

Analytic continuation and functional equations

Langlands proved that $E(g,\phi,\lambda)$ has meromorphic continuation in the complex spectral parameter $\lambda$ . Its constant terms are finite sums of global intertwining operators. Relations among normalized intertwining operators give the Weyl-group functional equations.

In rank one, $\lambda$ is often written as a single complex variable $s$ ; for a higher-rank Levi it belongs to the complex dual of the real split-center space of $M$ .

Spectral role

Values of Eisenstein series generate the continuous automorphic spectrum. Residues at suitable poles can be square-integrable and generate the residual automorphic spectrum. Thus Eisenstein series organize the noncuspidal part of the automorphic spectral decomposition recursively from cuspidal data on Levi subgroups.

L-functions

Normalizing factors of global intertwining operators are built from ratios of automorphic $L$ -functions in many settings. Their analytic behavior can therefore imply meromorphic continuation and functional equations for those $L$ -functions. This is a method with hypotheses, not a universal consequence for every representation of every $L$ -group.

Relation to the letter

The letter recognizes Eisenstein series as a route from functorial representation data to analytic properties of Euler products. Modern theory separates the induced representation, the meromorphic family, its constant terms, and the resulting spectral constituents.

References

Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, 1976. DOI.
C. Mœglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995. DOI.