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.



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.


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