Multiplicities in Plancherel theorem for SL2(R)

Question

The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. But this theorem doesn't give any multiplicities. For instance, let $D$ be a discrete series representation of $G=SL_2({\mathbb R})$, then the $G\times G$ representation $D\otimes D^*$ is a subrepresentation of $L^2(G)$. What is its multiplicity? So what is the dimension of the space $$ \mathrm{Hom}_{G\times G}(D\otimes D^*,L^2(G))? $$ Likewise, what are the multiplicities of the continuous Hilbert integrals corresponding to the rest of the spectrum. I am looking for a paper which gives the $G\times G$ decomposition of $L^2(G)$.

Answer 1

These multiplicities are all 1. Abstract Plancherel theorem (in e.g. Dixmier’s C*-algebras, 18.8.1):

$L^2(G)=\int_{\hat G}^\oplus H_\pi\otimes H_\pi^*\,d\mu(\pi)$, for Plancherel measure $\mu$ on any type I loc. compact unimodular $G$.