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 here][1]https://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).

[Cohomology for quotient bundle][2]


  [1]: https://en.wikipedia.org/wiki/Kempf_vanishing_theorem
  [2]: https://mathoverflow.net/questions/263761/sheaf-cohomology-of-the-universal-sub-and-quotient-bundles-of-the-grassmannian?rq=1