# first cohomology of exterior power of tangent bundle

suppose G is a Grassmannian manifold, and TG is the tangent bundle.

By Bott's theorem $H^1(G, T_G)=0$.

Is it true that $H^1(G, \bigwedge^i T_G)=$, for i>0.

I saw some vanishing result by lepotier, but would like to confirm.

thanks.

All summands of $\Lambda^iT_G$ correspond to irreducible representations of the Levi group $L$ with $G$-dominant highest weights. Because of this their higher cohomology vanishes.