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Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation $$f(g)=\int_{\hat{G}_\mathrm{temp}}\mathrm{Trace}(\pi(\lambda(g)\check{f}))d\mu_p(\pi),$$ where $\check{f}(g)=f(g^{-1})$.

Can one provide a reference where I can find an explicit description of $d\mu_p(\pi)$?

Only known case to me is for $\mathrm{SL}_2(\mathbb{R})$, where $$d\mu_p(\pi)= \begin{cases} & r\tanh(\pi r)dr, \text{ when $\pi$ is the principal series representation with parameter $ir$}\\ & nd_{\mathrm{count}}, \text{ when $\pi$ is the discrete series representation with highest weight $-n-1$}. \end{cases}$$ I wish to have a similar description in case of $G$. Thanks in advance.

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    $\begingroup$ The plancharel measure has been explicitly computed by Harish-Chandra (at-least for spherical vectors) and it is involved with what's known as HC "c" function. $\endgroup$
    – Asaf
    Commented Sep 17, 2016 at 17:21
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    $\begingroup$ Yes this is known to me. But why should that formula, which is for the symmetric spaces, extend to the group case? $\endgroup$
    – Subhajit Jana
    Commented Sep 17, 2016 at 17:40
  • $\begingroup$ arxiv.org/pdf/1101.3753.pdf $\endgroup$
    – user1688
    Commented Sep 18, 2016 at 6:40

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