Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence $B$ acts on $R$. Then one can construct an induced $G$-algebra as follow:
$$ S=(k[G]\otimes R)^B. $$
Now, is it true that the spectrum of $S$ is the associated scheme $G\times^BX$? 



spectrum of an induced algebra
If $X=pt$, then $G\times^B X=G/B$ is the flag variety, which is not affine. 


However we have $S=\Gamma(G\times^B X, \mathcal O)$ and hence $spec(S)$ is the affinization of $G\times^B X$.


