Next semester I may study a course where the ultimate goal is to get to the Borel – Weil – Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have studied some representation theory of Lie groups from Brian Hall's book Lie groups, Lie algebras and Representations. I am ok with highest weight theory for $\mathfrak{sl}_n$ and I have also studied some Schur – Weyl duality and classification of irreps of $\mathfrak{sl}_n$ from Fulton and Harris.
My question is: What would a learning roadmap for understanding the BWB Theorem be?
I was told by the lecturer we would probably start out by looking at line bundles over $\Bbb{P}^1$. Now I don't mind if there is no single reference/book to read linearly that I have to look into. Though, I don't know how one would build up one's background to get to the theorem.
I don't mind if I have to learn things like sheaf cohomology and the like on the way.
If it helps, I have also studied differential geometry and am familiar with the material in chapters 1-5,7,8,11,14,16 of Lee's Smooth Manifolds, second edition.
Best Answer
The two, broadly defined things you need to know are Lie theory and (complex) differential geometry. The specific things from each topic are
Lie groups/algebras
Highest weight theory of compact Lie groups/complex semisimple Lie groups. One needs to build up to the theorem that irreducible complex representations of a Lie algebra are parametrized by dominant positive weights.
Most books (including Brain Hall's) build up to this. I personally like Compact Lie Groups by Sepanski if one already understand the basics of manifolds, as its very concise, has good exercises, and concludes with a proof of the Borel-Weil theorem.
Differential geometry
One needs to know what a holomorphic vector bundle over a complex manifold is and their Dolbeault cohomology. Thus the following general differential geometry background is needed, all of which the last can be found in Lee's Smooth Manifolds
Vector bundles over a manifold (I think Lee mostly talks about real vector bundles but complex ones are just replacing $\mathbb R$ with $\mathbb C$).
Differential forms.
After having a solid differential geometry background, in my opinion the best place to learn the necessary complex geometry is part 1 of the freely available notes by Moroianu, titled Lectures on Kahler Geometry.
For the Borel-Weil-Bott theorem proper, I would see Sepanski's text and also the paper Representations in Dolbeault Cohomology by Zierau.