From the [Peter-Weyl theorem in wikipedia][1], 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][2], 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**


  [1]: http://en.wikipedia.org/wiki/Peter-Weyl_theorem
  [2]: http://www.springerlink.com/content/xp7xvqu6ahncmf4v/