Infinitesimal automorphisms and deformations of parabolic geometries

We show that infinitesimal automorphisms and infinitesimal deformations ofparabolic geometries can be nicely described in terms of the twisted de-Rhamsequence associated to a certain linear connection on the adjoint tractorbundle. For regular normal geometries, this description can be related to theunderlying geometric structure using the machinery of BGG sequences. In thelocally flat case, this leads to a deformation complex, which generalizes theis well know complex for locally conformally flat manifolds. Recently, a theory of subcomplexes in BGG sequences has been developed. Thisapplies to certain types of torsion free parabolic geometries including,quaternionic structures, quaternionic contact structures and CR structures. Weshow that for these structures one of the subcomplexes in the adjoint BGGsequence leads (even in the curved case) to a complex governing deformations inthe subcategory of torsion free geometries. For quaternionic structures, thisdeformation complex is elliptic.