I am interested what kind of (different) proofs of meromorphic continuation Eisenstein series (for general parabolic subgroups) exist in the literature? The only one I understand well, is Bernstein's proof using his "continuation principle" (it is probably unpublished) but apparently there are many others (especially I am interested in proofs which are different from the one in Langlands book)
A belated response: so far as I can tell, Selberg's idea, taken literally, does not apply at all in rational rank above 1. One should note that Avakumovic and Roelcke had similar ideas, which also did not anticipate the complications of higher rank. Langlands' 544 (notes written in the mid60s, not public until mid 70s) were extremely novel in their recognition of complications in higher rank, e.g., cuspidaldata Eisenstein series in the first place, and nonconstant residual automorphic forms (e.g., Speh forms). ColindeVerdiere's argument works well in rank 1, but, in its nascent form, has the same limitations as Selberg's 1950s viewpoint. MoeglinWaldspurger cite Langlands and others, but do not give proofs of several analytical points. Bernstein's apocryphal proofs of meromorphic continuation are reputed to be instantiated last fall... but one should not be overoptimistic, given the possibility of people finding other priorities. Around 2001, I tried to rewrite notes on Bernstein's idea, with help from notes obtained thanks to Hejhal and Sarnak. I think it is fair to say that there are several confusing points, even if other potentially confusing points can be cleared up by "standard mathematics". In the last few years, there has been interest in supposedly applying Bernstein's method [sic] to notcuspidaldate Eisenstein series... My personal reaction, based on some experience, is skeptical. I would like to see (and may try to write it myself) actual proofs for cuspidaldata Eisenstein series. :) A significant caution is that various spaces of automorphic forms meeting growth conditions are not representation spaces for the relevant group G, so reasoning that implicitly assumes this is dangerous. Of course, one often needs less... Lisa Carbone and Howard Garland have recently written some things about Eisenstein series on notclassicalgroups (e.g., looporsomething)... that seem to succeed, based aesthetically/morally on the BernsteinSelberg arguments. If anyone wants further technical information about my assessment of the situation, I welcome email about meromorphic continuation of Eisenstein series. :) Edit (15 April '12): [By the way, deleted my last year's comment on misspelling in the question, which I could not correct last year...!] Disregarding the relatively special cases where arguments based on Poisson summation can succeed, as far as I can tell, all other approaches need some compactness or finitedimension assertion at some crucial juncture. (The usual way to prove some space is finitedimensional is to exhibit it as a nonzero eigenspace for a compact operator!) Granting some such assertion, the remainders of the arguments are relatively formal. E.g., inside Colin de Verdiere's argument is an essential Rellichlike compactness assertion about a resolvent, in a form due to FaddeevPavlov and LaxPhillips. I suspect that the essential "confusing" or "mysterious" aspects of proofsketches reside in problems about this sort of point. Not that incorrect conclusions are reached, but that the complexity of the setup often gives an impression that one has done sufficient work to have completed a/the proof, and "surely" an "auxiliary" finitedimensionality statement oughtn't be critical? That is, it's not that various "confusing" argument are incorrect, but, rather, perhaps incomplete. Correctibly so, indeed, but perhaps not trivially. 


I'm not familiar enough with the proofs to say if they are more than superficially different, but here is something: MoeglinWaldspurger prove continuation crediting Jacquet (see p. xix), instead of Langlands. They say it is similar to that given by Efrat, in his treatment of the Hilbertmodular ($PSL_2$ over a totally real field) case. Here, Jacquet credits MW's proof to Colin de Verdière's "new and strikingly brief and elegant proof" (from the MR review) for $SL_2({\mathbb R})$ (extended here). I couldn't find any extensions of Colin de Verdière's argument to higherrank cases, but that may be because it was done in MoeglinWaldspurger. Muller also has a proof in the rankone case. Wong gave a proof using integral equations. Everyone listed credits Selberg with their ideas. 


You'll find proofs of analytic continuation (and functional equation, both generally come together) in Weil's "Basic number theory", Bump's "Automorphic forms and representations", in Godement's "Séries d'Eisenstein" (available here), in Hida's "Elementary theory of $L$functions and Eisenstein series", Garrett's "Holomorphic Hilbert modular forms" and a host of other places... 


I think that in particular Erez Lapid has done a nice job with these two slides http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid1.pdf http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid2.pdf Have a look in particular on page 10 in the first slide session for Bernstein's prinicple, and a proof of it is on page 11. He has also states it in for $SL(2)$ on page 9, which is always helpful for me before seeing a general statement. The second slides focus on the higher rank situation. By the way, the whole side is great: http://www.math.clemson.edu/~jimlb/coursenotes.html Perhaps a remark about Eisenstein series: I think that at least in a congruence setting, the analytic continuation is in some sense equivalent to analytic continuation of automorphic L function. The LanglandsShahidi http://en.wikipedia.org/wiki/LanglandsShahidi_method method deduces from the analytic continuation of Eisenstein series the analytic continuation of automorphic $L$ functions. So every new proof of analytic continuation for Eisenstein series yields a new proof for the analytic continuation of automorphic $L$ functions. 

