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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.

Rushabh Mehta
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2 Answers2

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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.

  • de Rham theory.
  • Connections on vector bundles (maybe not necessary but I think it's useful).

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.

Eric O. Korman
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  • Thanks for your answer. I went to the library today to borrow Sepanski's book on Lie groups. I will also read up on vector bundles from my copy of Lee's book. –  Jun 07 '13 at 15:32
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Question: "I was told by the lecturer we would probably start out by looking at line bundles over P1. 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."

Answer: If $k$ is the complex numbers and $V:=k\{e_0,e_1\}$ and $V^*:=\{x_0,x_1\},$ it follows $V$ and $V^*$ are irreducible $G:=SL(2,k)$ modules. Here $G$ is the group of $2 \times 2$-matrices with coefficients in $k$ with determinant $1$.

You may check the following: If $C:=Proj(k[x_0,x_1])$ is the complex projective line and $\mathcal{O}(d)$ is the $d$'th tautological linebundle on $C$ ($d\in \mathbb{Z}$), it follows the global sections $H^0(C, \mathcal{O}(d)) \cong Sym_k^d(V^*)$ is (when $d\geq 1$) the $d$-th symmetric product of $V^*$ - which is an irreducible finite dimensional $G$-module. Using this construction you may realize all irreducible finite dimensional $G$-modules as global sections of invertible sheaves on $C$. Here we realize $C$ as a quotient

$$C\cong SL(2,k)/P,$$

where $P \subseteq SL(2,k)$ is a parabolic subgroup. Hence there is a canonical action of $SL(2,k)$ on $C$ inducing an action on the global sections of $\mathcal{O}(d)$. The Borel-Weil-Bott theorem generalize this correspondence to an arbitrary semisimple group. You find an elementary exposition of this in the Fulton/Harris book in the GTM series.

Example: If $G$ is a semi simple algebraic group over $k$ and $V(\lambda)$ is an irreducible finite dimensional $G$-module, there is a parabolic subgroup $P \subseteq G$ (not unique) and an invertible sheaf $L(\lambda) \in Pic(G/P)$ with an isomorphism of $G$-modules

$$ V(\lambda) \cong H^{l(w)}(G/P, L(\lambda)).$$

Here $G/P$ is a smooth projective variety of finite type over $k$ (a flag variety) and $H^{l(w)}(G/P, L(\lambda))$ is sheaf cohomology of the linebundle $L(\lambda)$.

In algebra/geometry the quotient construction is complicated:

Quotient of a Lie algebra by a subalgebra - what is it?

hm2020
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