A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg Lie Algebras

In this paper we construct a holomorphic polynomial first order differential representation of the Lie algebra which is the semidirect sum hn o sp(2n,R), on the manifold Cn × Dn, different from the extended metaplectic representation [6]. The case n = 1 corresponding to the Lie algebra h1 o su(1, 1) was considered in [3]. The natural framework of such an approach is furnished by the so called coherent state (CS)-groups, and the semi-direct product of the Heisenberg-Weyl group with the symplectic group is an important example of a mixed group of this type [11]. We use Perelomov’s coherent state aproach [12]. Previous results concern the hermitian symmetric spaces [2] and semisimple Lie groups which admit CS-orbits [4]. The case of the symplectic group was previously investigated in [1], [6], [5], [10], [12]. Due to lack of space we do not give here the proofs, but in general the technique is the same as in [3], where also more references are given. More details and the connection of the present results with the squeezed states [13] will be discussed elsewhere.