| In this thesis we define a new notion of rank for unitary representations of semisimple groups. The definition is based on studying the restriction of the representation to certain unipotent subgroups of semisimple groups which are expressible as towers of extensions by Heisenberg groups. Kirillov's classical method of orbits for nilpotent groups is the main tool in our definition.;The construction of such unipotent subgroups is totally based on root systems and therefore is generel; i.e. it applies to both classical and exceptional groups at the same time. The main point of this thesis is to prove that the notion of rank provides a nontrivial finite-valued invariant associated to a representation. We also study the universal enveloping algebras associated with the unipotent groups which were mentioned above, and in particular we show that the centers of these algebras are indeed polynomial algebras. |
