# 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 $sl_3$?

Thanks in advance

• 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

## 1 Answer

For the last question [edit: this part of the question has now been removed..], the relation of constructible sheaves with the Fukaya category is the subject of the paper Microlocal branes are constructible sheaves by David Nadler, and its predecessor Constructible Sheaves and the Fukaya Category by Nadler and Eric Zaslow. As for D-modules in general, there are proposals in the physics dictionary, starting in work of Anton Kapustin (see e.g. here) and perhaps best summarized in the seminal paper of Kapustin-Witten. One has to be careful though what exactly you mean: to get nonholonomic D-modules, the idea is to look at coisotropic but non-Lagrangian A-branes, ie not the Fukaya category but an enlarged version that has yet to be defined mathematically. Also even in the Lagrangian case there are two really different things one can mean by Fukaya category of the cotangent bundle, depending on how you treat behavior at infinity. Nadler uses a refined version that corresponds precisely to constructible sheaves. The one that arises in the physics, and in most of the math literature (eg work of Abouzaid) is the "wrapped Fukaya category" -- this corresponds more closely to modules over infinite order differential operators - in particular delta functions at distinct points on the base become identified (by exponentiating translation)! so this is quite far from what you might want in say representation theory.

As to your other question, I'm not sure I understand the issue -- D-modules form a perfectly nice stack, i.e. they glue together, independently of holonomicity (one doesn't need Riemann-Hilbert to see this, it's immediate from the definition as modules over a sheaf of algebras - in fact from my, perhaps naive, point of view it's easier to see perverse sheaves form a stack by thinking of them as D-modules, but that's certainly not necessary either). So the gluing formalism you ask for is just sheaf theory if I understand correctly: D-modules on a cover plus gluing data define a D-module on the total space. Same holds on the (dg) derived category level. This is in fact how you even define what a D-module on a stack is (in terms of smooth covers), as is discussed at great length in the last "chapter" of the text by Beilinson-Drinfeld on Quantization of Hitchin's Hamiltonians.

Now for the question of describing simple modules, that's a seriously tricky issue that I know nothing about -- I believe Toby Stafford was the first to show that there even are simple but nonholonomic modules over rings of differential operators, and I would look at his papers for insight.

• @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
• Well at least if you have a simple D-module, it has to be simple locally, right? (a nontrivial sub globally has to be cosupported somewhere..) – David Ben-Zvi Jun 23 '10 at 17:22
• 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