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 mid-60s, not public until mid 70s) were extremely novel in their recognition of complications in higher rank, e.g., cuspidal-data Eisenstein series in the first place, and non-constant residual automorphic forms (e.g., Speh forms). Colin-de-Verdiere's argument works well in rank 1, but, in its nascent form, has the same limitations as Selberg's 1950s viewpoint. Moeglin-Waldspurger 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 over-optimistic, 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 not-cuspidal-date 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 cuspidal-data 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 not-classical-groups (e.g., loop-or-something)... that seem to succeed, based aesthetically/morally on the Bernstein-Selberg arguments.

If anyone wants further technical information about my assessment of the situation, I welcome email about meromorphic continuation of Eisenstein series. :)

Best to all,  -paul

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 mid-60s, not public until mid 70s) were extremely novel in their recognition of complications in higher rank, e.g., cuspidal-data Eisenstein series in the first place, and non-constant residual automorphic forms (e.g., Speh forms). Colin-de-Verdiere's argument works well in rank 1, but, in its nascent form, has the same limitations as Selberg's 1950s viewpoint. Moeglin-Waldspurger 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 over-optimistic, 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 not-cuspidal-date 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 cuspidal-data 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 not-classical-groups (e.g., loop-or-something)... that seem to succeed, based aesthetically/morally on the Bernstein-Selberg 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 mis-spelling 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 finite-dimension assertion at some crucial juncture. (The usual way to prove some space is finite-dimensional is to exhibit it as a non-zero 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 Rellich-like compactness assertion about a resolvent, in a form due to Faddeev-Pavlov and Lax-Phillips. I suspect that the essential "confusing" or "mysterious" aspects of proof-sketches reside in problems about this sort of point. Not that incorrect conclusions are reached, but that the complexity of the set-up often gives an impression that one has done sufficient work to have completed a/the proof, and "surely" an "auxiliary" finite-dimensionality 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.