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**.