TITLE: Borel's presentation for the cohomology of a Flag Variety
QUESTION [8 upvotes]: If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then 
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^W$ is a polynomial ring,
where $K$ is a field of characteristic zero. I found this in page 2 here.  
Q: For which $G$ does 1) and 2) continue to hold if $K$ has positive characteristic? 
I'm guessing the answer is that the statements hold in positive characteristic if $G$ is special as an algebraic group, but I'm having trouble finding references in general, since I don't know this theory. It would also be nice to know if the story is the same if I replace $G$ with a split reductive group and replace $H^*$ with Chow $A^*$, or if the setup works more generally if we replace $K[T^{\vee}]^W$ with $H_G^{*}(\ast)$ (but I was confused by the not generated in degree 2 comment in Knutson's answer here). 
Originally posted on mathstackexchange, but I was worried asking about characteristic assumptions might be too specific there.

REPLY [9 votes]: The answer to your question is contained in Demazure's paper: https://eudml.org/doc/142233
In particular, as you noted, the group is special iff the Borel presentation holds in any charactertistic.
If you want to know the structure in characteristic $p$, there is another useful reference by Victor Kac: https://link.springer.com/article/10.1007%2FBF01388548