the primitive degree terms, a detailed comparison with the computation for the dense subset in [@BF14c] implies that the inclusion must be an isomorphism.

For each $x \in W_{\mathrm{af}}$ and $\lambda \in P$, we have an associated $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-equivariant line bundle ${\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}(\lambda)$ over ${\mathbf{Q}_{G}}(x)$. Also, for each $x \in W$ and $\lambda \in P^+$, we have a Demazure submodule $V^{-}_{x} ( \lambda )$, in the sense of [@Kas05], of the level-zero extremal weight module $V ( \lambda )$ (of extremal weight $\lambda$) over the quantum affine algebra $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ associated to ${\mathfrak{g}}_{{\mathrm{af}}}$.

\[fBWB\] For each $x \in W_{\mathrm{af}}$ and $\lambda \in P$, we have $${\mathop{\rm gch}\nolimits}H^{i} ( {\mathbf{Q}_{G}}( x ), {\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}( \lambda ) ) = 
  \begin{cases} 
    {\mathop{\rm gch}\nolimits}V^{-}_{x} ( - w_{\circ} \lambda ) & \text{\rm if $i = 0$ and $\lambda \in P^{+}$}, \\[1.5mm]
    0 & \text{\rm otherwise},   \end{cases}$$ where ${\mathop{\rm gch}\nolimits}$ denotes the character taking values in $({\mathbb{Z}}{(\hspace{-1pt}(q^{-1})\hspace{-1pt})})[P]$, and $w_{\circ} \in W$ is the longest element.

The higher cohomology vanishing part of Theorem \[fBWB\] is based on the fact that the ring $R_G$ is free over a polynomial ring with infinitely many variables (Theorem \[R-free\]), which is also an interesting result in its own. We should mention that Theorem \[fBWB\] have 