Does any one know what the correct formulation of the plancherel theorem should be for Homogeneous spaces. More specific I am looking for a statement like: there is a unique measure in $\mu$ in $\hat G $ such that $L^2(G/H)=\int_{\hat G}^{\oplus}H(\xi)d\mu(\xi)$ and something like a functional $I(f)=\int_{\hat G}^{\oplus}Tr(\xi(f))d\mu $ I will appreciate a lot your help. I am more familiar with the language of C^* algebras so if you can state this in that setting will be even better.
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$\begingroup$ What restrictions are you typically imposing on G and H? It isn't clear to me what such a formula would mean if G were the free group on two generators and H were the identity; but that could just be my ignorance. $\endgroup$ – Yemon Choi Sep 10 '11 at 3:56

1$\begingroup$ G is a reductive algebraic group over a local field. And H is a closed subgroup. From this you can conclude that G is a postliminal separable group. And there is hope for a plancherel measure. In the case you stated the free group in two generators is not separable so I am not sure we can say something there. $\endgroup$ – Carlos De la Mora Sep 14 '11 at 5:42

$\begingroup$ Thanks for the clarification. "The free group in two generators is not separable"  in what sense do you mean "separable"? $\endgroup$ – Yemon Choi Sep 14 '11 at 20:27

$\begingroup$ It does not have a dense countable subset. $\endgroup$ – Carlos De la Mora Sep 15 '11 at 18:40

1$\begingroup$ Carlos, have you seen the book "Lie Theory: Harmonic Analysis on Symmetric SpacesGeneral Plancherel Theorems"? A version of van den Ban's contribution is available on his website (under Lecture Notes). He also has a survey in PSPM 61 ("Harmonic Analysis on semisimple symmetric spaces. A survey of some general results.") which is available from his website (under Publications). $\endgroup$ – B R Sep 21 '11 at 5:19
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May be the paper: "MR0444844 (56 #3191) Penney, Richard Abstract Plancherel theorems and a Frobenius reciprocity theorem. J. Functional Analysis 18 (1975), 177–190" is what you are looking for.