Collineations of polar spaces with restricted displacements

Abstract

Let J be a set of types of subspaces of a polar space. A collineation (which is a type-preserving automorphism) of a polar space is called J-domestic if it maps no flag of type J to an opposite one. In this paper we investigate certain J-domestic collineations of polar spaces. We describe in detail the fixed point structures of collineations that are i-domestic and at the same time (i + 1)-domestic, for all suitable types i. We also show that {point, line}-domestic collineations are either point-domestic or line-domestic, and then we nail down the structure of the fixed elements of point-domestic collineations and of line-domestic collineations. We also show that {i, i + 1}-domestic collineations are either i-domestic or (i + 1)-domestic (under the assumption that i + 1 is not the type of the maximal subspaces if i is even). For polar spaces of rank 3, we obtain a full classification of all chamber-domestic collineations. All our results hold in the general case (finite or infinite) and generalize the full classification of all domestic collineations of polar spaces of rank 2 performed in Temmermans et al. (to appear in Ann Comb).