Kostant theorem on Lie algebra cohomology of nilpotent subalgebra

introduction

• one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},M(\mu))$ 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

thm (Kostant)

Let $\lambda\in \Lambda^{+}$. For a finite dimensional highest weight representation $L({\lambda})$ of a complex semi-simple Lie algebra $\mathfrak{g}$ $$ H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} $$

related items

• BGG resolution
• Bott-Borel-Weil Theorem
• Lie algebra cohomology

expositions

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