That'* what I'm talking about. If there'* play in the pulley, and it'* as you describe, the coupler is either extremely and dangerously worn, or already broken and nonexistent.
Stop driving the car immediately.
Read my supercharger page to understand it a bit better:
The DRIVE sequence of a setup like this is described rather simply: Belt drives pulley. Pulley spins input (nosedrive) shaft, . Behind the pulley is a seal to prevent Supercharger oil from escaping. Nosedrive shaft spins in two bearings lubricated by this oil. At the end of this shaft are 3 pins that lock into a flexible coupler (this coupler prevents gear damage by absorbing the shock load of a quick jump on the throttle, so the force flexes the coupler rather than break gear teeth). This coupler joins the input shaft to one of the rotor shafts via 3 identical pins. So when the input shaft spins, one rotor spins. Each rotor has an identical gear on it with a 1:1 ratio. So the two rotors spin in a counter-rotating fashion. One clockwise, the other counter-clockwise. The rotors shafts are sealed between the gear housing and compressor chamber for two reasons. One reason is to prevent manifold pressure (boost) from pressurizing the oil chamber (causing leaks) and the other is to prevent the oil from leaking into the compressor housing. Each rotor shaft is retained by a bearing on each end of the shaft.
The 3 pins on the gear can be clearly seen in this pic, and one of the mating pins in the nosedrive is also visible. There should be ZERO play in that pulley. The coupler is there to join them together.