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In Dixmier ($C^*$-algebras), the Plancherel theorem states (I will not mention the right regular representation even though the theorem does talk about it):

Let $G$ be unimodular, $\lambda$ be the left regular representation of $G$, $\mathcal{U}$ be the Von Neumann algebra on $L^2(G)$ generated by $\lambda(G)$, $J$ the mapping $f \mapsto f^*$ on $L^2(G)$, and $t$ the natural trace on $\mathcal{U}^+$ defined by $\epsilon_1$. Then there exists a positive measure $\mu$ on $\hat{G}$ and an isomorphism $W$ of $L^2(G)$ onto $\int^\oplus (K(\zeta) \otimes \overline{K(\zeta)}) d\mu(\zeta)$ (I`m guessing the integral is over $\hat{G}$) with the following properties:

($i$) $W$ transforms

$J$ into $\int^\oplus J_\zeta \ d\mu(\zeta)$

$\lambda$ into $\int^\oplus (\zeta \otimes 1) \ d\mu(\zeta)$

$\mathcal{U}$ into $\int^\oplus (\mathcal{L}(K(\zeta)) \otimes \mathbb{C}) \ d\mu(\zeta)$

$t$ into $\int^\oplus t_\zeta \ d\mu(\zeta)$

etc...

However, it seems that in the case $G$ is a real reductive Lie group this decomposition can be refined where the decomposition is done over parabolic subgroups $P = MAN$, discrete series of $M$ and unitary induction on elements of $i\mathfrak{a}^*$. According to my colleague it should look something like this (in naive terms):

$L^2(G) = \bigoplus_{P = MAN} \bigoplus_{\mu \in M^\wedge_{ds}} \int_{\nu \in i\mathfrak{a}^*} \pi_{P, \mu, \nu} d\xi$

where $M^\wedge_{ds}$ is the set of discrete series reprensetations of $M$, $d\xi$ is some measure related to Harish-Chandra`s $c$-functions, and $\pi_{P, \mu, \nu}$ is some class of representations of $G$ depending on these parameters.

My main question is this: I am looking for references on this refinement, preferably very self-contained references.

My second question is, in N. Wallach`s book Real Reductive Groups II, he proves a very close thing, namely the decomposition of $L^2(G/K)$ where $K$ is a maximal compact subgroup obtained by a Cartan involution, and the decomposition of $L^2(G/N, \chi)$, where $N$ is the nilpotent radical of a parabolic subgroup $P = MAN$ and $\chi$ is a character of $N$. Can one obtain the refined decomposition of $L^2(G)$ with the above two decompositions in some way?

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  • $\begingroup$ That's Harish-Chandra's PLancherel-Theorem and it can be found in his three part paper. You also might want to look at the 1986 paper of Herb and Wolf. Also Knapp's book is a good source... $\endgroup$ – user1688 Jan 30 '18 at 13:19
  • $\begingroup$ Is the three part paper you`re talking about called Harmonic analysis on real reductive groups? $\endgroup$ – Henrique Tyrrell Jan 30 '18 at 13:24
  • $\begingroup$ Yes, that's the one. $\endgroup$ – user1688 Jan 30 '18 at 18:55

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