## The Plancherel Formula for SL2

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What are the important questions to start to study about Plancherel Formula for SL2? And what is the relation this topic and the Arthur's paper, "On some problems suggested by the trace formula (1984)" Can anyone suggest any source shows this relation?

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## 2 Answers

Another standard reference for the Plancherel formula is the book: $SL_2(\mathbb{R})$, Addison-Wesley, 1974; of course $SL$ stands for Serge Lang.

The Plancherel formula and the Selberg trace formula have in common that a trace is computed in two ways. But they deal with different representations of $G$. For $G$ a suitable locally compact group (type I, unimodular, separable), and $f\in C_c(G)$, the Plancherel formula is $f(e)=\int_{\hat{G}}Tr\pi(f)\;d\mu(\pi)$, where $\mu$ is the Plancherel measure on the dual $\hat{G}$; and the difficulty is, in concrete examples, to describe $\hat{G}$ and $\mu$ explicitly (or at least to describe explicitly the support of $\mu$, called the tempered dual).

For the Selberg trace formula, I found the Wikipedia article fairly readable: http://en.wikipedia.org/wiki/Selberg_trace_formula

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I can't speak to the second question, but a canonical source on Plancherel for $SL(2)$ is

An Introduction to Harmonic Analysis on Semisimple Lie Groups V.S. Varadarajan

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