Kostant theorem on Lie algebra cohomology of nilpotent subalgebra

introduction

• one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},V(\mu))$ is $\mathbb{C}_{\mu}$ for $i=0$ for $i>0$.
• this requires knowing the BGG resolution, which is a stronger result since it carries information about homomorphisms between Verma modules

expositions

• http://www.math.columbia.edu/~woit/LieGroups-2012/borelweilbott.pdf