I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties. 
[![Klyachko - Equivariant vector bundles on toric varieteis][1]][1]

Whereas detailed literature regarding the constructiob of Klychko's filtration is available (e.g. [Perling][2]) I was not able to find references discussing the part on cohomology. I have the following questions:
1. Is anyone aware of references that discuss and prove the part on cohomology?
2. Does theorem 3.1 also work for reflexive sheaves?
3. Consider $X=\mathbb{P}^n$ and the cotagnet bundle $\Omega_{\mathbb{P}^n}$. For this case, [we know][3] that the filtrations are given by
[![enter image description here][4]][4]
In particular, consider $n=2$ and the complete fan with rays $\rho_0 =(1,0), \rho_1 = (0,1), \rho_2 =(-1,-1)$ with the maximal cones $\sigma_0 =Cone(\rho_0,\rho_1), \sigma_1 =Cone(\rho_1,\rho_2), \sigma_2 =Cone(\rho_2,\rho_0)$. How do we compute the space $E_{\sigma}(\chi)$ and the complex $C^{*}(E,\chi)$ for a given cone $\sigma$ and a given character $\chi$?  


  [1]: https://i.sstatic.net/rE8PD.jpg
  [2]: https://arxiv.org/abs/math/0205311
  [3]: https://arxiv.org/abs/1910.13964
  [4]: https://i.sstatic.net/NBCGb.jpg