Exceptional extended dual polar spaces (Q1372619)

It is considered an extended dual polar space (EDPS). An EDPS \({\mathcal G}\) is called affine if there is a flag-transitive automorphism group \(G\) and a normal subgroup \(T\) in \(G\) such that \(T\) acts regularly on the set of elements of type 1 in \({\mathcal G}\). Theorem. Let \({\mathcal G}\) be an EDPS whose rank is greater than or equal to 3 and which possesses a flag-transitive automorphism group with finite stabilizer of maximal flag. Then one of the following holds: (1) \({\mathcal G}\) is isomorphic to one of the 19 exceptional EDPSs (there are 10 geometries of rank 3, 8 geometries of rank 4 and 1 geometry of rank 5); (2) there is a 2-covering \(\widetilde {\mathcal G} \to {\mathcal G}\), where \(\widetilde {\mathcal G}\) is affine.