PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2095–2102 S 0002-9939(98)04223-3 LIE INCIDENCE SYSTEMS FROM PROJECTIVE VARIETIES ARJEH M. COHEN AND BRUCE N. COOPERSTEIN (Communicated by Ronald M. Solomon) Abstract. The homogeneous space G/P λ , where G is a simple algebraic group and P λ a parabolic subgroup corresponding to a fundamental weight λ (with respect to a fixed Borel subgroup B of G in P λ ), is known in at least two settings. On the one hand, it is a projective variety, embedded in the pro- jective space corresponding to the representation with highest weight λ. On the other hand, in synthetic geometry, G/P λ is furnished with certain subsets, called lines, of the form gB〈r〉P λ /P λ where r is a preimage in G of the funda- mental reflection corresponding to λ and g ∈ G. The result is called the Lie incidence structure on G/P λ . 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. 1. Introduction Let k be a field, n a natural number, and let X be any set of points in the projective space P(k n ). The points and lines lying on X give rise to an incidence system which we shall denote by Δ(X ). We are interested in the structure of these incidence systems. The main motivating examples stem from the geometries of Lie type. In these cases, the set X is actually a projective variety, given as an intersection of quadrics. 1.1. Theorem (cf. [2], [8]). Suppose G is a connected split semisimple algebraic group with Tits system (B,N,W,R), and set T = B∩N . Let λ be a dominant weight with respect to B, and let P λ = BW λ B be the corresponding parabolic subgroup of G. Then the G-module S 2 V (λ) contains the highest weight module V (2λ) ∗ with multiplicity 1 and with a G-invariant complement M . The highest weight orbit G〈v λ 〉 of G in P(V (λ)) is a projective variety isomorphic to G/P λ . It is the zero set of any basis of M (which are homogeneous quadratic polynomials on V (λ)). This result is probably well known. A proof for k of characteristic 0 can be found in [2], and a more general proof in [8]. In the setting of the above theorem, let λ be a fundamental node, say λ = ω j , and denote the corresponding reflection by r. Then, according to the theorem, the point set of each Lie incidence system can be identified with the zero set of all (quadratic) polynomials in M . Here, we recall from [3], the Lie incidence system associated with G and r is a pair Γ = (P , L), where P = G/P λ and L consists of all Received by the editors July 6, 1996 and, in revised form, December 18, 1996. 1991 Mathematics Subject Classification. Primary 51B25; Secondary 14L17, 14M15. Key words and phrases. Groups of Lie type, Lie incidence systems, geometry, quadrics. c 1998 American Mathematical Society 2095 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use