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.
Best Answer
The Borel subgroup of the quantized function algebra $G_q$ is of course certain quotient Hopf algebra $p:\mathcal{O}(G_q)\to \mathcal{O}(B_q)$ which then canonically coacts on $G_q$ from both sides (from the left by $(p\otimes id)\circ\Delta_{G_q}:\mathcal{O}(G_q)\to \mathcal{O}(B_q)\otimes \mathcal{O}(G_q)$. In the case of $SL_q(n)$ you get the lower Borel by quotienting by the Hopf ideal generate by the entries of the matrxi $T$ of the generators standing above the diagonal.
Of course, if you define $Qcoh(G_q)$ as the category ${}_{\mathcal{O}(G_q)}\mathcal{M}$ of left modules over $\mathcal{O}(G_q)$ then the category of relative left-right Hopf modules ${}_{\mathcal{O}(G_q)}\mathcal{M}^{\mathcal{O}(B_q)}$ is precisely the category of quasicoherent sheaves over the quantum flag variety $G_q/B_q$. If you look at several of my earlier articles they together show that in the case of $SL_q(n)$ at least, there is a collection of localizations of the above category of relative Hopf modules which are affine, which gives it a structure of a noncommutative scheme with a canonical atlas whose cardinality is the cardinality of the Weyl group (the generalization to other parabolics is easy). Moreover the forgetful functor from the category of Hopf modules to ${}_{\mathcal{O}(G_q)}\mathcal{M}$ is the inverse image part of the geometric morphism which is a Zariski locally trivial $B_q$-fibration; the local triviality, which is obtained with the help of the quantum Gauss decomposition, boils down to the Schneider's equivalence for the extension of the algebras of localized coinvariants into the corresponding localization of $G_q$. The sketch of the whole program is in
For the Peter-Weyl look at the original works of S. Woronowicz or