Lie incidence systems from projective varieties

Abstract

The homogeneous space G / P λ G/P_{\lambda } , where G G is a simple algebraic group and P λ P_{\lambda } a parabolic subgroup corresponding to a fundamental weight λ \lambda (with respect to a fixed Borel subgroup B B of G G in P λ 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 λ \lambda . On the other hand, in synthetic geometry, G / P λ G/P_{\lambda } is furnished with certain subsets, called lines, of the form g B ⟨ r ⟩ P λ / P λ gB\langle r\rangle P_{\lambda }/P_{\lambda } where r r is a preimage in G G of the fundamental reflection corresponding to λ \lambda and g ∈ G g\in G . The result is called the Lie incidence structure on G / P λ 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.