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## Automorphic forms on GL(3)

We know that the classical Maass forms on GL(3) are depicted, for instance, in D.Goldfeld's book. I wonder that if there exists "holomorphic" automorphic forms on GL(3) as an analogue of GL(2) case. If those forms exist, where can I find the materials which concretely tell the story of them?

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The symmetric space attached to GL(3) is not a complex manifold, so no strict analogue of "holomorphic" is available. Perhaps the right analogues are "algebraic" or "cohomological", which is satisfied by (for example) symmetric square liftings of holomorphic GL(2) forms. There are also non-self-dual forms of this type which are not lifts. – David Hansen Oct 24 2010 at 13:46
As David knows, both "algebraic" and "cohomological" are close, but not the same, as "holomorphic" for GL_2, even for cuspidal auto reps. Algebraic auto reps for GL_2 include certain non-holomorphic auto reps (those attached to continuous even irreducible Galois reps) and and cohomological ones correspond to weight k>=2 forms, so don't see weight 1. "Discrete series" is another good buzz-word---but unfortunately I don't think GL_3 admits any discrete series at all. So it's very difficult to see a good analogy. – Kevin Buzzard Oct 24 2010 at 16:46
Some comments on discrete series: A theorem of Harish-Chandra is a (linear connected semisimple Lie) group has discrete series if and only if the rank of the group is equal to the rank of the maximal compact subgroup (which is equivalent to the existence of a compact Cartan subgroup). By comparing ranks of SL_n and SO(n), you see that discrete series only exist there for n=2. Further, not all discrete series are holomorphic, and when a group does have holomorphic discrete series, it can still have non-holomorphic discrete series (like real symplectic groups). – BR Oct 24 2010 at 21:07
AFAIK, the "classical" way to generalize holomorphic automorphic forms from GL(2) is to look Siegel modular forms on Sp(2n,R), since SL(2,R) is isomorphic to Sp(2,R). And, of course, Hilbert modular forms, which, if you don't know, live on GL(2,F) where F is a totally real number field. – BR Oct 24 2010 at 21:23

As in the comments and earlier answer: in short, there is nothing comparably elementary or accessible for GL(3), as holomorphic things for GL(2).

Even the explication of this apparent fact is not, and perhaps could not be, as immediate as the direct exhibiting of holomorphic things for GL(2): to demonstrate the absence of things is harder than showing presence.

The most "elementary" GL(3) forms might be the Gelbart-Jacquet lifts from GL(2), although what is readily describable is not the automorphic form(s) but the L-functions. In any case, this is a "global" description which, therefore, is inevitably more complicated.

A "local" description of (unitary?) repns of the GL(3,R) versus GL(2,R) is much easier than global comparisons. GL(2,R) basically has holomorphic discrete series and principal series, the former (globally) giving holomorphic automorphic forms, the latter giving waveforms. The repn theory of GL(3,R) includes no discrete series (this is not obvious). It does include both (irreducible) principal series and repns induced from the holomorphic discrete series on the copy of GL(2) in the 2,1-block or 1,2-block parabolic subgroup(s). In the last few decades, the "cohomological" repns have been distinguished, although this is less easy to describe than principal series or induced repns generally.

In practice, none of these repn types are as accessible as the holomorphic things for GL(2) on the upper half-plane.

The older thread in which holomorphic automorphic forms were the exclusive topic was Siegel modular forms, indeed. To compare the repn theory, this was possible because Sp(n,R) (or "2n"...) does have holomorphic discrete series repns, for all n. Siegel and H. Braun did not address things in such terms, and Harish-Chandra's early work on repn theory seems to have overlooked the implicit appearance of holomorphic discrete series in Siegel's work!

Sp(n,R) (2n-by-2n matrices) in fact has $n$ different discrete series. For example, the 4-by-4 case has the holomorphic ones, and what are called the "big" discrete series. The repn theory also includes all induced repns from Levi components of parabolics. Thus, in fact, the relative poverty of unitary repns of GL(n,R)'s gives them a greater simplicity than the Sp(n,R)'s, despite the fact that the story is not elementary.

Certainly holomorphic discrete series are among cohomological ones, and I believe (Clozel more-or-less confirmed this a few years ago) that at least case-by-case Vogan and his students have shown that all discrete series are cohomological.

It is also considerably surprising that the old (1960s) results of Matsushima and Murakami on which automorphic forms can appear in cohomology really does show that it depends only on the archimedean representation type.

So we probably must reconcile ourselves to "cohomological" being the "right" generalization of "holomorphic".

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 Thanks for your comment! Very glad to meet you here:) -xin wang – Alex Jun 29 2011 at 12:55

You can see that there are no holomorphic analogues to the Maass forms in Goldfeld's book because those functions depend on 5 real variables: $y_1, y_2, x_1, x_2, x_3$. Any function of (one or more) complex variables must be a function of an even number of real variables.

This is just a lowbrow version of David Hansen's assertion that the symmetric space is not a complex manifold.

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