When a Lie group G has a closed normal subgroup N, the "Mackey Machine" breaks down the classification of its irreducible representations into two smaller problems: a) find the irreducible representations of N; b) find the irreducible projective representations of certain subgroups  of G/N. The desired classification often follows inductively.  Key parts of this machine are 1) the "inducing construction" (building representations of G out of those of its subgroups); 2) the "imprimitivity theorem" (characterizing the range of the  inducing construction); 3) a "tensoring" construction (combining objects of types a) and b) above). Many years ago Kazhdan, Kostant and Sternberg defined the notion of  inducing a Hamiltonian action from a Lie subgroup, thus introducing a  purely symplectic geometrical analog of 1); and the question arose whether analogs of 2) and 3) could be found and built into an effective "symplectic Mackey Machine." In this talk I will describe a complete solution to this problem, obtained recently.