Knowlpedia

Eisenstein Series on a Reductive Group

A series induced from a parabolic whose analytic continuation produces -functions

Let G/FG/F be reductive and P=MNP=MN a parabolic subgroup with Levi MM.

Given a (suitably KK-finite) section fsf_s in a representation induced from P(AF)P(\mathbb A_F), the Eisenstein series is

E(g,f,s)=∑γ∈P(F)\G(F)fs(γg),E(g,f,s)=\sum_{\gamma\in P(F)\backslash G(F)} f_s(\gamma g),

convergent for Re(s)≫0\mathrm{Re}(s)\gg 0.

Remarks

Key properties (Langlands):

  • E(g,f,s)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,π,ϕ)L(s,\pi,\phi) should continue meromorphically and sometimes satisfy functional equations.