TITLE: Is there analogue of Peter-Weyl theorem for non-compact or quantum group
QUESTION [7 upvotes]: From the Peter-Weyl theorem in wikipedia, this theorem applies for compact group. I wonder whether there is a non-compact version for this theorem.
I suspect it because the proof of the Peter-Weyl theorem heavily depends on the compactness of Lie group. It is related to the spectral decomposition of compact operators. 
Thanks Mariano pointing out the Peter-Weyl theorem does not hold for non-compact group. But I really wants to know is: is there any Peter-Weyl analogue decomposition for non-compact group, say decompose to integral representations but not finite dimensional representations?
Another related questions is about the definition of quantized flag variety. In the work of Lunts and Rosenberg on localization for quantum group, they tried to establish the quantum analogue of Beilinson-Bernstein localization theorem. They defined the quantized flag variety in the framework of noncommutative algebraic geometry. They used the Peter-Weyl philosophy for quantum group to define the coordinate ring of quantized base affine space as the direct sum of all simple $U_{q}(g)$-modules with highest weight $\lambda$(positive).(Denoted by $R_{+}$)
Then one can define category of quasi coherent sheaves on "quantized flag variety" as proj-category of graded $R_{+}$. 
What I want to ask is there any other way to define quantized flag variety? In the classical case, It is well known that flag variety can be define as $G/B$, say $G$ is general linear group and $B$ is Borel subgroup. Is there any analogue for quantum case? Is there a definition like $G_{q}$ as "quantum linear group" and $B_{q}$ as quantum analogue of Borel subgroup?
However,I suspected, because the quantum flag variety is essentially not a real space

REPLY [6 votes]: Marty's answer discusses the Plancherel formula, and in a comment on his answer, I mentioned Harish-Chandra's work on the Plancherel formula in the case of reductive Lie groups. Yemon Choi's answer also mentions the case of semisimple Lie groups as being easier than the general case.  The point of this answer is to elaborate slightly on my comment, and to point out that, while the semi-simple case might be easier, it is a very substantial piece of mathematics; indeed, it is essentially Harish-Chandra's life's work.
When Harish-Chandra began his work, it was known (thanks to Mautner?) that semisimple Lie groups were type I, and hence that for such a group $G$, the space $L^2(G)$ admits a
well-defined direct integral decomposition into irreducibles (typically infinite
dimensional, if $G$ is not compact).  However, this is a far cry from knowing the precise
form of the decomposition.  
The most fundamental, and difficult, question, turns out to
be whether there are any atoms in the Plancherel measure, i.e. whether $L^2(G)$ contains
any non-zero irreducible subspaces, i.e. whether the group $G$ admits discrete series.
This was solved by Harish-Chandra, who established his famous criterion: $G$ admits discrete series if and only if it contains a compact Cartan subgroup.  He also gave a complete enumeration of the discrete series representations up to isomorphism, and described
their characters via formulas analogous to the Weyl character formula.
Harish-Chandra then want on to describe
the Plancherel measure on $L^2(G)$ inductively in terms of direct integrals
of parabolic inductions of discrete series represenations of Levi subgroups of $G$.
(It is the appearance of Levi subgroups, which are always reductive but typically
never semisimple, that also forces one to generalize from the semisimple to the reductive case.)
After completing the theory of the Plancherel measure for reductive Lie groups, he then went on to develop the analogous theory for $p$-adic reductive groups.  However, in this case, one still doesn't have a complete enumeration of the discrete series representations in general: there are certain "atomic" discrete series representations, called "supercuspidal", which have no analog for Lie groups, and which aren't yet classified in general (i.e. for all $p$-adic reductive groups). 
Harish-Chandra's work, as well as standing on its own as an amazing edifice, was a central
inspiration for Langlands in his development of the Langlands program, and remains 
at the core of the Langlands proram today.
For a very nice introduction to Harish-Chandra's work, and the surrounding circle of ideas,
one can read this article by Rebecca Herb.