Lie Incidence Systems from Projective Varieties

The homogeneous space $G/P_{\lambda }$, where $G$ is a simple algebraic group and $P_{\lambda }$ a parabolic subgroup corresponding to a fundamental weight $\lambda$ (with respect to a fixed Borel subgroup $B$ of $G$ in $P_{\lambda }$), is known in at least two settings. On the one hand, it is a projective variety, embedded in the projective space corresponding to the representation with highest weight $\lambda$. On the other hand, in synthetic geometry, $G/P_{\lambda }$ is furnished with certain subsets, called lines, of the form $gB\langle r\rangle P_{\lambda }/P_{\lambda }$ where $r$ is a preimage in $G$ of the fundamental reflection corresponding to $\lambda$ and $g\in G$. The result is called the Lie incidence structure on $G/P_{\lambda }$. The lines are projective lines in the projective embedding. In this paper we investigate to what extent the projective variety data determines the Lie incidence structure.