Borel Weil Bott theorem without complex analysis?

Question

Is it possible to prove the Borel-Weil-Bott theorem without learning complex analysis? Is it recommended?

I know sheaf cohomology, representation theory, vector bundles and some tools of Lie group theory, but I do not know much about holomorphic bundles. Can I learn these without learning some serious complex analysis?

Reason: I find the complex analysis stuff that I see in textbooks very boring and I find myself working through it extremely slowly in comparison to other more familiar topics - I don't know if I have the time to really trudge through it in regard to my study planning.

If I do indeed need to learn complex analysis, what would I need? I need to know, in order to determine if I have time in the near future to even look at this.

Answer 1

I don't think that you need much complex analysis to prove the Borel-Bott-Weil theorem. If you use the approach via Kostant's theorem, then to deduce BBW you need the Peter-Weyl theorem and the fact that you can compute the coholomolgy of the sheaf of holomorphic sections of a vector bundle using Dolbeault cohomology. Kostant's theorem itself (which describes the Lie algebra cohomology of the nilradical of a parabolic subalgebra of a semisimple Lie algebra) is purely algebraic. The vector bundles showing up in the formulation of BBW and its proof basically all are homogeneous vector bundles, so you can get holomorphic bundles and holomorphic sections via holomorphic representations and holomorphic equivariant functions, respectively.

Both the proof of Kostant's theorem and how to derive BBW from it can be found in Kostant's 1961 article "Lie algebra cohomology and the generalized Borel-Weil theorem" (MR 26#265).