if the load of an inverter(using a battery) is a pure reactive load, will the battery be drained ( neglect the efficiency of DC/AC conversion)?
If you neglect efficiency, there will be no drain on the battery with an inductive reactive load. An inverter will not work very well with a capacitive reactive load. To supply a reactive load, an inverter must have an internal capacitor that is large enough to handle the reactive current. Since there is no such thing as a lossless inverter, the losses will need to be supplied by the battery. The losses in the inverter depend on the voltage and current ratings and design details. Losses could be less than 5 percent but are likely to be twice that or more. There will also be some losses in any reactive load.
The answer depends on the exact circuit used in the inverter. Most likely the DC power draw will be less than for a resistive AC load of the same current. Either find a spec in a datasheet that tells you what the inverter does with reactive loads, or just measure it.
Don't expect the measurements from one model to apply to other models.
It would be a good idea to have a chat with a field application engineer from the inverter company.