A321 flight crew reported just after leveling off at cruise altitude they got an ECAM warning message for number 1 engine hydraulic pump low pressure.

Narrative

Just after level off at cruise altitude we received an ECAM warning message; HYD G ENG 1 PUMP LO PR. My Aircraft" We conducted numerous checklists. The PTU (Power Transfer Unit) was pressurizing the green hydraulic system. I noticed we probably could reach ZZZ Maintenance via radio and it was a strong reception. This problem was explained to them and they contacted Maintenance Control. Numerous ACARS messages were sent to Dispatch and in return received messages. I and Dispatch determined it would be a good idea to divert to ZZZ1. Numerous calls were made to the flight attendants and passengers to keep them informed. I [requested priority handling]. Safety vehicles took a look at the aircraft on the runway. We had the safety vehicles follow us to the ramp. I requested a tow onto the gate. Did not want to be close to any equipment under aircraft power. Safety representatives came aboard along with maintenance personal to discuss the situation. Cause: Our simulator instructors prepare us well for this type of events. "My Aircraft " This is so important to clarify throughout the event. The person flying and the person that is performing tasks."