Lie incidence systems from projective varieties

The homogeneous space G/P-lambda, where G is a simple algebraic group and Pr 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[r]P-lambda/P-lambda where r is a preimage in G of the fundamental reflection corresponding to lambda and g is an element of 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.