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Multicontact mappings on Hessenberg manifolds

This thesis was inspired by work of M. Cowling, F. De Mari, A. Koranyi and M. Reimann, who studied multicontact structures for the homogeneous manifolds G/P, where G is a semisimple Lie group and P is the minimal parabolic subgroup of G. The multicontact structure here arises naturally by the nilpotent component N of the Iwasawa decomposition of G, which is an open and dense subset of G/P. In the thesis two problems are addressed. The first one concerns the representation of simple Lie algebras in terms of polynomial algebras. More precisely, a polynomial basis for split simple Lie algebras is explicitly given in a suitable choice of coordinates. The second problem investigates the multicontact structure on Hessenberg manifolds. These manifolds can be locally viewed as quotient of the nilpotent component in the Iwasawa decomposition of semisimple Lie groups. Then the multicontact structure is inherited by the one in the Iwasawa nilpotent case. Rigidity with respect to this structure is studied and proved for a large subclass of Hessenberg manifolds.