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The question is contained in the title. I would guess that this question is already answered in the literature.

Given the reductive group $GL(n)$ over a complete local field, how does the right regular representation on $L^2(G(F))$ or perhaps better on $L^2(G(F)/Z(F))$ or $L^2(G(F)^1)$ decompose, where $Z$ is the center of $G$. Even less intuitive, how does the right regular decomposition on $C_{c}^{\infty}$-version or the Harish-Chandra Schwartz space look?

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up vote 10 down vote accepted

For the Lie (a.k.a. "archimedean") case, interpreting the $L^2$ question as asking for Plancherel measure, Harish-Chandra in-principle did this for a large class of reductive groups. The early non-compact example was $GL_n(\mathbb C)$ treated by Gelfand and Naimark, where the orbital-integral idea already appeared, in a much simpler guise since there's only one conjugacy class of maximal torus. Evidently they thought that all unitary irreducibles should appear in $L^2(G/Z)$, so that it remained to the 1960s for Knapp-Stein to find the unitary degenerate unitary principal series. Effectively because of the "patching" issue between conjugacy classes, the "real" case, even just $GL(n,\mathbb R)$, is in later papers of Harish-Chandra. Knapp's Princeton book is an easy reference, or Harish-Chandra's collected works (which are out of print?)

In-principle, Harish-Chandra and Arthur further decomposed Schwartz functions, for reductive Lie $G$. Arthur did address compactly-supported and Paley-Wiener space, but in a different style than the question, I think.

For p-adic $G$, as commented in Silberger's Princeton notes, Harish-Chandra had a Plancherel theorem for $L^2$ and corresponding for Schwartz spaces in 1971-73, although Silberger's notes stop short of developing what was known at the time, perhaps because there was no reasonable parametrization of supercuspidals at the time. By now, with Bushnell-Kutzko and Gross-Reeder et alia, in-principle things could be assembled.

The two articles by Moeglin in the Proc AMS 61 Edinburgh conference from 1996 address repns of $GL(n,\mathbb R)$ and $GL(n,\mathbb Q_p)$, with references, but not Plancherel.

But I can't point to a tangible source for unadorned Plancherel for $GL(n)$'s. The Lie case can be obtained by specializing the general cases (the only discrete series on Levis are the holomorphic discrete series on the GL(2)'s). The p-adic case? Someone else?

Edit: Looking at the Edinburgh 1996 volume, I see that Helgason has a piece in which he recaps G-N's and HC's only-one-conjugacy-class (of maximal tori) argument for "discovering" the Plancherel formula, in the Lie case. (To me, this issue of "discovery" is very important.) He also gives the first case where there's the patching issue, $SL_2(\mathbb R)$.

Edit 2: in response to comment/further-question: the parametrization of the "spectrum" is qualitatively very similar to the automorphic $L^2$ spectral decomposition, but also qualitatively simpler. Comparing the Lie and automorphic cases roughly: in the Lie case, the "bulk" of $L^2$ is spherical unitary principal series, with parabolic-induced repns from discrete series on Levis also entering, as well as actual discrete series, if any, on the whole group. In the automorphic case, the "bulk" of $L^2$ is spherical cuspidal repns, by Weyl's Law [sic]; in many situations, provably (and conjecturally) the bulk are unitary principal series. (In general, a naive form of Ramanujan-Petersson conjectures cannot be true, for a variety of reasons... e.g., lifts, as in Howe-PiatetskiShapiro, Corvallis, AMS Proc Sympt 33, 1977/79, ... but Weyl's Law stuff still says that probably most are unitary principal series.) The parabolically induced (Eisenstein series) from afms on Levis is a smaller part of the spectrum.

That is, in broad terms, parabolic induction from special objects on the Levis plays a completely analogous role. (This played a role in some otherwise mysterious choices of terminology in the local repn theory, e.g., "Eisenstein integrals" and "Maass-Selberg relations" in HarishChandra.)

In both cases, there are "atoms" which are to some degree mysterious: locally, discrete series, globally, cuspforms. There are certainly global issues that have much-more-trivial local counterpart, such as Weyl's Law stuff or Ramanujan-Petersson.

Yet again, much more can be said, and there is much literature on details. Not sooo many sources on higher-rank.

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Thanks for this answer. Lot of material here to be checked. But you don't know anything compromised as one can find e.g. in Gelbart-Jacquet for the global picture $L^2( GL(2, k) \backslash GL(2, \mathbb{A})=$ cuspidal rep. + characters + direct integrals involving parabolic induction? – Marc Palm Jul 20 '11 at 17:55
Thanks for also answering my consecutive answer. I have to diggest the information, you gave, but that seems to describe the situation pretty thorough. – Marc Palm Jul 20 '11 at 21:38

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