On the local recognition of finite metasymplectic spaces

Abstract

This paper is concerned with the local recognition of certain graphs and geometries associated with exceptional groups of Lie type. The local approach to geometries is inspired by group theory. Finite simple groups are often characterized by local information, for example, the fusion pattern of involutions centralizing a given involution. The main results here, although of a geometric nature, are a contribution to obtaining a characterization of a group of exceptional Lie type by the fusion pattern of root subgroups centralizing a given root subgroup. Let d be the shadow space of a (thin or thick) building of spherical type M,, (where n indicates the rank) with respect to a given node r of A4 (cf. Tits [ 13, 143). We shall view d as a space, i.e., as a set of points together with a collection of subsets of size at least two of the point set, called lines. Thus, the points of A are the vertices of type r of the building in question and the lines are the residues of flags of cotype {r}. The local recognitions we intend to discuss are based on the fact that up to (nonspecial) isomorphisms, the building is uniquely determined by d. If p is a point of A, the se1 A, r(p) of points collinear with p (including p) constitutes a subspace in the sense that each line bearing two distinct points of A,,(p) is entirely contained in A Gl(p). A space with this property is often called a gamma space. Since A affords a group of automorphisms which is transitive on the point set if n > 2 (cf. [ 13]), we can associate with A a space A ,c, such that for each point p of A the subspace A .,(p) of A is isomorphic to A G r. (Here and elsewhere, a subspace X is regarded as a space by taking 348 0021-8693/89 $3.00