Symplectic polarities of buildings of type E 6

A symplectic polarity of a building ∆ of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality θ of ∆ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E6 of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E6 for which the Phan geometry is empty.