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 here).

Originally posted on mathstackexchange, but I was worried asking about characteristic assumptions might be too specific there.

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.