$D$-affinity of Quadrics Revisited

Let $K$ be aa algebraically closed field of characteristic $p\ge 3$ and let $Q_n\subset {\mathbb{P}}_K^{n+1}$ be a smooth quadric hypersurface. We show that if $n=2m\ge 4$ then $Q_n$ is not $D$-affine. In particular, we show the grassmannian $Gr(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\ge 3$. Our result complements previous work of A. Langer, who showed that if $p\ge n=2m+1$ then $Q_n$ is $D$-affine.