Jed, that is the flexible coupler between the motor and the pump. It drives the pump pulley at full motor speed, while allows the clutch to slowly come up to speed via friction for driving the transmission pulley.
and FYI, the one in the photo has been hastily cut. Apparently they didn't have a 3/8" nut driver to loosen the clamp to allow the other section of this part free. So they just cut it off.