Hooking a charger to a charged, rested battery will, typically, start a charge cycle. A generic explanation is that the float voltage put out by a charger is about 13.8 volts. Some smart chargers will put out up to 14.2 volts when charging a dissipated battery as a way of getting charge in quickly. After taking a battery off charge, the battery terminal voltage will drop a little as the surface charge (technical term) on the internal bits (non technical term) dissipate the surface charge until the battery reaches a rested state (technical term), probably in the 12.6 to 12.8 region. Individual setups vary a lot in those quoted numbers.
When you connect a charger to a rested battery, it will typically see a battery terminal voltage that is lower than the charger's idea of what is fully charged and start a charge cycle, on that basis. If the battery is, in fact, charged, the charge cycle will terminate quite quickly, indicated by the charger going in float. The forgoing may explain why you saw the charger indicate charging was happening.
Inverters often consume a "standby" current just by being connected to a battery. That current is typically quite low but will drain somewhat over time. A large inverter that is turned on using some deliberate switch operation, but not supplying an appliance, can consume upwards of an Amp just sitting there ready to go.
My answer to your original question is that you may be seeing normal indications of charger operation or the inverter is using some standby current and is draining the battery.
