# Is simple non-holonomic D-module a local concept?

It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a stack. Therefore, for example. If one has flag variety of $sl_2$,i.e. $P^1$, given its big cells as open affine covers: i.e. two $A^1$, then one can use this formalism to glue holomomic $A_1$-modules to holonomic $D-mod_{P^1}$,where $A_1$ is first Weyl algebra.

We know Description of simple holonomic D-modules on quasi compact schemes is local.

# My question

Is there any machinery that can glue non-holonomic simple D-modules from open affine covers to simple non-holonomic D-modules on total space? Is there a functorial way which can give a fairly complete list of simple non-holonomic $D-mod_{G/B}$. For example, let's consider the flag variety of $sl_3$; the big cells are given by 2-dimensional affine space $A^2$. Then we know there are a lot of non-holonomic $A_2$-modules(simple) from Bernstein-Lunts. In this case, how can we globalize them to non-holonomic D-modules on flag variety of $sl_3$?

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The references for your "Another question", I would guess, are the papers of Nadler--Zaslow and Nadler relating constructible sheaves on $X$ to the Fukaya category on $T^*X$; you can find them on the ArXiv. – Emerton Jun 23 '10 at 13:57
Shizhuo- I really would recommend sticking to one question at a time. It improves the possibility of getting good answers. Also, the papers Matt mentions are also mentioned in David's comment. – Ben Webster Jun 23 '10 at 15:07
@Ben, thank you for recommendation. I will delete the another question part and try to stick only one question in one post next time – Shizhuo Zhang Jun 23 '10 at 15:12

@David: Thank you for clarifying. Yes, I am sorry for misleading. What you talked about is correct: $U|\rightarrow D-mod_{U}$ is a stack. So one can globalize them from affine covers with gluing data. What I really want to know is the description of simple non-holonomic D-modules on the total space. For example, if I know the simple $A_n$-modules(non-holonomic), can I obtain the simple non-holonomic D-modules on the total space? – Shizhuo Zhang Jun 23 '10 at 15:28
Just a remark: Stafford's original counterexample to the belief that simple D-modules are holonomic involved proving that a specific differential operator generated a maximal left ideal in ${\mathcal D}$. There's a nice paper by Bernstein and Lunts, Invent. Math. 94 (1988), that shows that Stafford's original construction is actually not very sensitive to this choice of operator: roughly, a generic operator generates a maximal left ideal. – Thomas Nevins Jun 23 '10 at 18:36