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Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm looking for references that would be suitable as a follow up to James Humphrey's "Linear Algebraic Groups" and a first year graduate Algebraic Geometry course. I'd also prefer the texts to be in English but if it's necessary, I could also read references in French.

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3 Answers 3

I would encourage you to consider "Representation Theory and Complex Geometry" by Chriss and Ginzburg. In particular, I think you might enjoy the realization of irreducible representations of the Weyl group of a complex semisimple group $G$ on the Borel-Moore homology of the fibres of the Springer resolution of the nilpotent cone. For me, this has always been one of the motivating examples in geometric representation theory.

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Thank you for the reference. I've checked out a copy of the book and it looks interesting, particularly the part you mentioned. –  Siddharth Venkatesh Mar 16 at 4:42

Additionally to Peter Crooks answer I would recommend to study the book of Hotta and others : D-Modules, Perverse Sheaves, and Representation Theory

Here you can learn about derived categories and perverse sheaves/d-modules (which are essential tools to study geometric representation theory) and how they are connected to representation theory.

From here on it is not far to understand the geometry involved in the context of Kazhdan-Lusztig theory, Koszul Duality (in the sense of Beillinson, Ginzburg and Soergel), the Geometric Satake equivalence (Mirkovic-Vilonen).

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Thank you. I've been looking particularly for a reference for perverse sheaves and D-Modules. I will check this book out. –  Siddharth Venkatesh Mar 16 at 4:46

I am currently reading though the two books already mentioned (Representation Theory and Complex Geometry by Chriss and Ginzburg and D-Modules, Perverse Sheaves, and Representation Theory by Hotta et al) and I definitely recommend them. Two other references that I have found helpful are:

  • Fulton's Young Tableaux. This presents a snapshot of the geometry associated to $ GL_n $ (e.g. producing irreducible representations from lines bundles on flag varieties). It is also a relatively easy read!

  • Humphreys' Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Whilst this is not so much about the geometric side of the picture, what I got out of it is an appreciation of the types of questions people in this area try to answer.

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I have been using Humphreys' book recently because there's a course at UC Berkeley I'm taking which is currently going through the BGG resolution and Beilinson-Bernstein and Humphreys' text is the only one I've found which gives a good construction of the former. Thanks for the Fulton reference. I wouldn't have thought to look through that judging by the title alone. –  Siddharth Venkatesh Mar 16 at 4:45

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