Eisenstein Series on a Reductive Group

Let $G/F$ be reductive and $P=MN$ a parabolic subgroup with Levi $M$.

Given a (suitably $K$-finite) section $f_s$ in a representation induced from $P(\mathbb A_F)$, the Eisenstein series is $E(g,f,s)=\sum_{\gamma\in P(F)\backslash G(F)} f_s(\gamma g),$ convergent for $\mathrm{Re}(s)\gg 0$.

Key properties (Langlands):

• $E(g,f,s)$ admits meromorphic continuation and satisfies functional equations governed by intertwining operators.

In the letter: Eisenstein series motivate why the Euler products $L(s,\pi,\phi)$ should continue meromorphically and sometimes satisfy functional equations.