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:
-
What are the most important/must-know results in the field so far? (even just buzzwords like "BB-lcalization" can be helpful).
-
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"?).
-
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.
Best Answer
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:
Computations to do:
Buzz-People:
The lists are in the order that I thought of them; the order is not significant.