D-modules

D-modules form the essense of the geometric (=automorphic side) of the Langlands correspondence. Formally, D-modules are easy to define; however, your understanding will be very shallow unless you know what D-modules really are, say on usual schemes. As you go deeper into the theory, you also understand why it is so attractive: D-modules are very rich!

Here are some suggestions as to the reading for the general theory of D-modules. They all cover more or less all the necessary material, so you can choose one that you like the most, and go with it.

• 1. Lectures by J. Bernstein

  2. Lectures by A. Braverman. (A reworked and corrected version by P. Etingof is available from his website.)

  3. Lectures by R. Bezrukavnikov

  4. Lectures by V. Ginzburg

  5. Lectures by D. Milicic

Here are two "official" books:

• • "Algebraic D-modules" by A. Borel et al.

  • "D-modules, Perverse sheaves and Representation Theory", by R. Hotta, K. Takeuchi and T. Tanisaki.

Once we have the theory of D-modules on schemes, we can extend it to arbitrary prestacks. This will be done in the (technically crucial) Talk 1.2. However, it may be a good idea to face it with some prior knowledge of how this is done. Here are some references:

• 1. For D-modules on prestacks: Sections 5 and 6 of Drinfeld-Gaitsgory

  2. For the crystalline approach see Gaitsgory-Rozenblyum

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