Affine Extensions of Generalized Polygons

We study extensions of generalized polygons by affine planes and obtain a geometric characterization of the natural seven-dimensional linear representation of a group of type G2and the eight-dimensional spin representation of a group of type B3. If k is a perfect field of even characteristic, then the natural seven-dimensional k -module of G2(k) is reducible and admits a six-dimensional quotient. For finite fields k of even characteristic we give a characterization of this module.