A note on isoparametric generalized quadrangles

Abstract

We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces of Clifford type to Tits buildings of type C2 derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point.