# Primer on Eisenstein series

My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.

In my continuing journey of entering the world of automorphic forms and the Langlands program, I keep finding a particular stumbling block to my understanding: the Eisenstein series and its meromorphic continuation.

As the Eisenstein series is very central to all things automorphic from Langlands-Shahidi, Rankin-Selberg integrals, the spectral decomposition as well as the many arithmetic applications, I find it disturbing when the properties of the series are referred to as a massive black box.

Now, I have read Bump's Automorphic Forms and Representations in which he proves the continuation and functional equation from the properties of the constant term (ie: of Hecke-Tate L-functions), but as I understand it (so... vaguely), this is backwards to the main thrust of Langlands work: Langlands proves the analytic properties of the Eisenstein series without reference to the L-functions, which allows him to port such properties to any L-function one can associate to an Eisenstein series.

For higher rank groups, I understand things are much more complicated (non-constant residual forms, several constant terms, cuspidal Eisenstein series, etc.).

As of now, I have no sense of where the functional equation comes from, nor do I have sense of why a tool, which I understand as global parabolic induction, is so intimately tied to Langlands L-functions.

My hope is that there exists a way to come to an understanding of Eisenstein series (their analytic properties and role) without the extremely daunting task of working through Moeglin-Waldspurger's Spectral Decomposition and Eisenstein series or Langlands' monograph (monolith?) On the Functional Equation satisfied by Eisenstein series.

So this is my question:

Is there an expository reference which deals with the general theory (spectral role and the proof of the meromorphic continuation/functional equation) of Eisenstein series for reductive groups?

If the answer is in the negative, then my follow-up question would be

How do experts in automorphic representations and related fields (I am very interested in different approaches) understand the properties of and role played by Eisenstein series?

Finally, I want to say that my primary interest in this question is the Eisenstein series on reductive groups (or perhaps their covers). However, if there is intuition to be gained by looking at geometric or loop group analogues, I would be very interested.

I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), nor that Moeglin-Waldspurger aim for maximum generality/encyclopedic-ness, ... Selberg's sketches were of an even more different time. My colleague D. Hejhal confirms my speculation that Selberg did not think about cuspidal-data Eisenstein series, insofar as there was no mention of any such thing in his papers.

As I have noted elsewhere, Bernstein's ideas about meromorphic continuation of Eisenstein series are not well-documented, or, perhaps, not documented. Others may disagree, but I think that the gossip about "non-cuspidal-data Eisenstein series" being treated in any sane fashion is unreasonable...

To finally respond to the question: I think there does not currently exist any friendly, yet actually proof-y, expository treatment of any sort of general case of meromorphic continuation of Eisenstein series. (More on this below.)

At the same time, many people know very well "what is expected of" Eisenstein series, what role they play in spectral theory, and so on.

Indeed, this is a poster-child for the fact that frequently the function of an idea/thing is much simpler than its construction/justification. (Precedents: the real numbers, ...)

F. Shahidi's relatively recent book with AMS does show the general function of Eisenstein series in a sort of Langlands-programme-y setting, and in that regard is an excellent model for "using Eisenstein series". Indeed, most people should look "forward" in this way rather than worrying about whether or not some foundational thing was "really proven"...

The more direct questions about meromorphic continuation of Eisenstein series... are harder to usefully answer, unfortunately. If it's any comfort, the perception that these things are ... "not clear"... is easily arguably correct. As in some other MO comments/answers of mine, by this year I think that the most believable/persuasive argument for meromorphic continuation (of Eis with cuspidal data... which, the point is, enter in an $$L^2$$ spectral decomposition with respect to Casimir) is a refined form of the Colin-de-Verdiere/Lax-Phillips/Faddeev-Pavlov "discretization" arguments. That is, disappointing to me as well, the Selberg and/or Selberg/Bernstein arguments, while excellent high-level technical heuristics, have stunningly high hidden technical costs. Probably Selberg did not think in such terms, and it is pointless for me to conjecture what Bernstein thinks is trivial or not. :)

While I myself do have some plans to write up examples about Eisenstein series and spectral decompositions... based substantially on my on-line notes and "vignettes", I think the best guide is thinking consciously of the distinction between "definition/proof-of-foundational-properties" and "role in ...". That is, the assertion of the $$L^2$$ spectral decomposition (as representations? as Casimir-eigenspaces? ...) can be formulated without knowing the details of the proofs of meromorphic continuation ... as interesting as they might be. :)

EDIT: it may be worthwhile to mention that I did eventually fairly-carefully write up a family of examples of how to meromorphically continue Eisenstein series... among other things. There is a link at http://www.math.umn.edu/~garrett/m/v/ to a (legal!) PDF of my book "Modern Analysis of Automorphic forms, by Example", published with Cambridge Univ Press. Among other goals, one was to give legitimate, genuine analytical grounding.

• I hasten to comment that E. Lapid has written many wise notes about Eisenstein series... and has noted some crazy subtleties... but/and there is no trivial resolution/outcome. – paul garrett May 8 '15 at 0:23
• Sorry to resurrect an older answer, but I was thinking on the answer this morning, and wondering if you could elaborate on 1) the rigidity theorems you reference in the first paragraph and their relevance, and 2) the discretization arguments mentioned towards the end. No exposition or article I have read on Eisenstein series have mentioned such topics, and I am afraid I don't know to what you are referring. – Spencer Leslie Aug 25 '15 at 10:27
• The "rigidity" is the Mostow-Margulis-etal theorems that assert that most co-finite-volume discrete subgroups of semi-simple real Lie groups are "arithmetic". Further, the "congruence subgroup problem"'s positive resolution for essentially all higher-rank groups is that mostly these discrete groups are "of congruence type", so p-adic and adelic ideas are relevant. Langlands was trying to preserve some generalities that turned out not to exist, to some degree. The "discretization"... [cont'd] – paul garrett Aug 25 '15 at 13:06
• ... [cont'd] is the trick (one way or another) of identifying (self-adjoint) compact operators in a "spectral decomposition", to use the discreteness of their spectrum. Y. Colin de Verdiere used the (Lax-Phillips/Faddeev-Pavlov) discrete decomposition of spaces of pseudo-cuspforms to prove meromorphic continuation of genuine Eisenstein series, by observing that after an elementary modification, they satisfy a differential equation whose resolvent is compact... For example, at math.umn.edu/~garrett/m/v/cdv_eis.pdf there is an explication of the latter. – paul garrett Aug 25 '15 at 13:12
• >As I have noted elsewhere, Bernstein's ideas about meromorphic continuation of Eisenstein series are not well-documented, or, perhaps, not documented. Do the five lectures linked here count? math.uchicago.edu/~mitya/langlands.html – David Feldman May 16 '16 at 5:14