On construction of maximal parabolic centralizers of root elements and the Weyl group of type F4 in E6(K) for fields K of characteristic 2

Abstract

The purpose of this paper is to give an elementary and explicit construction of maximal parabolic centralizers of root elements in the Chevalley group E6(K) and to show that the centralizer of a Seigel involution in the Weyl group W of type E6(K) is the Weyl group of type F4, or equivalently it is the stabilizer of a totaly singular line L in W, using properties of the generalized quadrangle (Ω,L) of type O6−(2).The construction here of the maximal parabolic centralizers of root elements in E6K for fields K of characteristic 2, is somewhat novel, beginning as it does, with root bases of Levi-type and root bases of unipotent type, giving an explicit construction of the group generated by these two types.