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I am looking for a comprehensible material covering Plancherel formula for $SL(n,\mathbb{R})$ and $SL(n,\mathbb{C})$. Of course, I wouldn't mind reading an explanation for general semisimple Lie groups.

(I am reading Varadarajan's Introduction to harmonic analysis on semisimple Lie groups and I'm a bit lost there.)

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Have you already gone through the SL(2,R) case? If not, Lang's book of that name may be helpful. (I'm assuming you want to go through the details. If you just want a general overview, you could try Arthur's "Harmonic Analysis and Group Representations" article in the Notices.) –  Kimball Nov 9 '10 at 14:42
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Incidentally, Arthur doesn't really deal with SL(n), but it's just a general introduction to the Plancherel formula, in a more general setting, though getting specific with certain examples like GL(n). –  Kimball Nov 9 '10 at 14:47
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3 Answers

up vote 3 down vote accepted

Knapp's book Representation theory of semisimple groups : an overview based on examples has a chapter on various versions of the Plancherel theorem. It is also done (very carefully, with all the details, I suppose) in volume 2 of Wallach's Real Reductive Groups.

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As Varadarajan does note, complex reductive groups (implicitly, with forgetful functor from C-groups to R-groups applied) have a vastly simpler spectral theory, already treated by Gelfand-et-al in the late 1940s. Apart from Gelfand's old papers, which I cannot read because (sadly) I do not read Russian, I think Varadarajan's book comes the closest in the contemporary "book" literature to explaining the vastly simpler case of complex (semi-simple) Lie groups. (Although some of Gelfand-et-alia's papers were translated, they have many very-specific internal references to other untranslated papers). Given Serge Lang's interest in simplicities, I was a bit disappointed that Lang-Jorgensen's books and memoirs on related matters do not give a separate treatment of the complex case, apart from a few comments that the recursion in the general case terminates in the complex case.

It is true that in that book Varadarajan's writing style is unusually (and, to me, pleasantly) "organic", which seems to create a difficulty for some readers. Nevertheless, if V's book is too difficult, I would wager that Knapp's or Wallach's books (while excellent!) would be nearly impossible, in regard to SL(n,R) or SL(n,C) for n>2.

One reason for this is that, especially in Knapp, the main overall goal is to set up Langlands' classification. This is in a different direction from Plancherel.

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Dym and McKean's Fourier Series and Integrals discusses the specific case of $SL(n,\mathbb{R})$ in its very last section. Although apparently not available online, the book is essentially self-contained.

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