Symplectic induction, prequantum induction, and prequantum multiplicities

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary G-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian G-spaces, which (as we show) unfortunately fails to mirror the situation where more than one G-module "quantizes"a given Hamiltonian G-space. This paper offers evidence that the situation is remedied by working in the category of prequantum G-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces and establish Frobenius reciprocity as well as the "induction in stages"property.