It might be a stupid question.


How to glue perverse sheaves? I am considering the following example. Flag variety of $sl\_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $Perv(P^1)$. The big cells of flag variety of $sl\_2$ give the affine cover for it. In this case, they should be two $A^1$. We can also consider category of perverse sheaves on $A^1$. My question is:


1. How to glue two pieces of perverse sheaves on 1-dimensional affine spaces to that of projective spaces? More general, is there any gluing machinery which can globalize the perverse sheaves? It seems that Beilinson had a paper talking about this, but what I preferred is some expository notes explaining with some examples.
2. Maybe I need to ask this in another question. How can one define perverse sheaves on noncommutative space. I am aware that there is a paper by Amnon Yekutieli, James J. Zhang talking about [perverse sheaves on noncommutative space](https://arxiv.org/abs/math/0211309 "Amnon Yekutieli, James J. Zhang. Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes. 2006. https://zbmath.org/?q=an:1137.14300"). However, what they considered was not really a noncommutative space from my understanding. (If I made mistake or bullshit, point out please). They consider quasi coherent sheaves of (not necessarily) commutative algebra on commutative scheme. Which does not fit my need. I am considering the following example: Quantized flag variety of $sl\_2$ i.e. $Proj(O\_q(G/N))$ in the sense of [Lunts–Rosenberg](https://doi.org/10.1007/s000290050044 "Lunts, V., Rosenberg, A. Localization for quantum groups. Sel. math., New ser. 5, 123 (1999). https://zbmath.org/?q=an:0928.16020") (see also [Erik Backelin and Kobi Kremnitzer](https://arxiv.org/abs/math/0401108v2 "Erik Backelin, Kobi Kremnizer. Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups. 2004. https://zbmath.org/?q=an:1165.17304") and [Tanisaki](https://arxiv.org/abs/math/0309349 "Toshiyuki Tanisaki. The Beilinson-Bernstein correspondence for quantized enveloping algebras. 2003. https://zbmath.org/?q=an:1082.17010")). It is a noncommutative scheme. I wonder whether James Zhang has also defined dualizing complexes for this case. **How to define category of perverse sheaves on quantized flag variety**?
The motivation for this question is I think there should be **quantum version of Riemann–Hilbert correspondence**. Which should describe the categorical equivalence:


$Perv(Proj(O\_q(G/N))$ and category of quantum holonomic D-modules on quantized flag variety.


At present, I have more interest to know the answer of question 1. Thank you!



Gluing perverse sheaves?
Beilinson's *How to glue perverse sheaves* explains how one can glue perverse sheaves on a variety from perverse sheaves on a closed subvariety and its open complement (assuming that the closed subvariety is the set of zeros of a regular function on the big variety).


Gluing perverse sheaves from an open covering is a very different problem. There is a paper by Kazhdan and Laumon *Gluing of perverse sheaves and discrete series representations*, devoted to the second problem. Further developments in this direction include the papers [arXiv:math/9811155](https://arxiv.org/abs/math/9811155 "Bezrukavnikov, Polishchuk: Gluing of perverse sheaves on the basic affine space") and [arXiv:math/0104114](https://arxiv.org/abs/math/0104114 "Bezrukavnikov, Braverman, Positselskii: Gluing of abelian categories and differential operators on the basic affine space"). The second one contains some discussion of the general categorical formalism.


