Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to calculate $H^∗(G(2,4)_k,\mathcal{E}(d))$.

I think it might be related to Kempf vanishing theorem but I do not see how.

enter link description herehttps://en.wikipedia.org/wiki/Kempf_vanishing_theorem

I have asked this question in Math Stack Exchange but did not get a response.

Another post partially answered this (without twisting tautological bundle).