CASIMIR-OPERATORS OF SEMIDIRECT PRODUCTS OF SEMISIMPLE WITH HEISENBERG GROUPS

Casimir operators for semidirect products of some semisimple groups wi th Heisenberg groups are computed. The analysis is carried out using d ual representations on Pock space, wherein the action of the semidirec t products are related to their dual groups, namely certain unitary, o rthogonal, and symplectic groups. The compact symplectic group chain i s also investigated; by passing to the complexification, groups 'betwe en' the symplectic groups are constructed, which are of the form of se midirect products of symplectic groups with Heisenberg groups.