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](http://www.ams.org/journals/tran/2011-363-01/S0002-9947-2010-05163-3/S0002-9947-2010-05163-3.pdf). 


**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](https://mathoverflow.net/questions/21651/cohomology-of-flag-varieties)). 


Originally posted on [mathstackexchange](https://math.stackexchange.com/questions/3138284/borels-presentation-for-the-cohomology-of-a-flag-variety), but I was worried asking about characteristic assumptions might be too specific there. 



Borel's presentation for the cohomology of a Flag Variety
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>


