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I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.

I'm becoming increasingly fascinated by stuff related to geometric representation theory ($D$-Modules, geometric quantization, Langlands & CFT).

It's fair to say I'm mainly drawn to the geometric stuff but recently after reading a bit about the classification of semisimple lie algebras, I discovered the beauty in the non-geometric aspects as well. Still I feel like my knowledge of the representation side is a lot more basic than the geometric side.

Thus, my question is three-fold:

  1. What are the most important/must-know results in the field so far? (even just buzzwords like "BB-lcalization" can be helpful).

  2. How much beyond the finite representations of finite groups should one know before diving into this subject? (Example: Is knowledge of the classification of real semisimple lie algebras a prerequiste for "D-Modules, Perverse Sheaves, and Representation Theory"?).

  3. What would be a good roadmap to geometric representation theory with an eye toward geometric Langlands? In particular one aimed at a geometer with a minimal representation theory background?

By a "geometer" I mean someone with enough background in Differential/Complex/Modern-Algebraic Geometry to find himself around the geometric side of the story while having only knowledge of representation of finite groups on the representation side.

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Since this has many answers, I'll just put down what comes to mind.

It seems like you would be content to not worry about real semisimple or other fields and just grant yourself use of $\mathbb{C}$ and of $\mathbb{C}((t))$. That would be the safest option to specialize everything to this.

So you've already read some classification of semisimple so I presume you have some Weyl character computations done.

Some ordinary representation theory:

  1. Why you are going to restrict yourself to category $\mathcal{O}$. How it is broken into blocks.
  2. Translation functors on above
  3. Classical Satake isomorphism using $\mathbb{C}((t))$. ( Note that wikipedia uses a stricter definition ). Pre-Geometric Satake.

Computations to do:

  1. Do some Birkhoff factorizations
  2. Trivialize some small rank vector bundle V on complex curve over the complement of finitely many points.
  3. D-modules on $\mathbb{P}^1$, get an example of BB. Write these differential operators and pick some $\mathcal{D}$ modules to see which reps they give.
  4. Some Bruhat Decompositions. Maybe do a non-type-A example. Parameterize a double bruhat cell as nicely as you can while here.

Buzz-People:

  1. Mirkovic-Vilonen
  2. Gan-Ginzburg
  3. Algebraic Peter-Weyl
  4. Higgs Bundle and Hitchin System
  5. Riemann-Hilbert
  6. Fourier-Mukai
  7. Deligne - Geometric Abelian Class Field Theory
  8. Kazhdan-Lusztig polynomials
  9. Braden-Licata-Proudfoot-Webster

The lists are in the order that I thought of them; the order is not significant.

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  • $\begingroup$ Wow, it's extremely helpful to have all these things written down so I can get back to this answer every now and again and check my progress. Although I'm fine with schemes at the level of hartshorne I don't feel very comfortable with the general theory of curves (their moduli space, picard schemes and jacobians), Is there a source you can recommend for this? I have the impression curves are all over this theory. $\endgroup$
    – Saal Hardali
    Commented Apr 24, 2016 at 15:27
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    $\begingroup$ @SaalHardali Do you have the complex intuition for them in terms of say Forster Riemann Surfaces? $\endgroup$
    – AHusain
    Commented Apr 24, 2016 at 22:01

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