Branching rules for restriction of the Weil representations of Sp(n,R) to its maximal parabolic subgroup CM(n)

The symplectic group Sp(n, R) is the group of linear canonical transformations of a real 2n-dimensional phase space and CM( n) subset of Sp(n, R) is a maximal parabolic subgroup. The symplectic groups are the fundamental dynamical groups of classical and quantal Hamiltonian mechanics. In particular, Sp(3,R) is the dynamical group of the spherical harmonic oscillator and its Weil (harmonic series) representations are important for the microscopic (shell model) description of the collective motions of many-particle systems. The subgroup CM( 3) subset of Sp(3,R) also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus. Thus, the Sp(3,R) --> CM(3) branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model. Some new developments are made in the vector-coherent-state theory of induced representations. (C) 1998 American Institute of Physics. [S0022-2488(98)01211- 0].