Borel-Weil from Sepanski´s book "Compact Lie Groups":
My intuition for Borel-Weil:
We have a compact Lie group G. And we want to “describe it nicely” - we want to find its representation to shed more light on G.
Borel-Weil describes one type of representation of G as holomorphic sections of complex line bundles. The representations that Borel-Weil gives are irreducible (which is an extra type of very suitable representations with certain advantages).
Then, Wikipedia says: The representations are realized in the global sections of holomorphic line bundles on the flag manifold of the group G. This just specifies how the representation is constructed, starting from the flag manifold on G.
The Borel-Weil is important, because without this, we wouldn't be able to describe the irreducible representations of compact Lie group G explicitly.
My question:
I fail to connect the theoretical description of what Borel-Weil does and its statement. Does the $V(w_0 \lambda)$ mean the representation of $G$ and the theorem states that for $\lambda$ dominant, it is isomorphic to the holomorphic sections? (On the left side)
I think my particular problem is understanding the role of weights and Weyl chamber. If we fix one weight $\lambda$, the next steps are constructing $V(w_0 \lambda)$ and then finding an isomorphism to the sections?
I would really appreciate seeing steps of the construction and what are we even constructing using this theorem?
Why I am asking I have seen an example for G = U(2), where “applying the Borel-Weil” is lenghty process on several pages, it works with a projective space $CP^1$ for some reason and ends with “Weyl Dimension Formula”. So before I try to understand all the steps, I would like to know the steps in general.
Thank you, hope this makes sense and my intuition is correct. Will also appreciate any comments on Borel-Weil, what you recommend to study etc.