I think, in typical legal system, institution authorized to examine whether was comitted a crime or not, is prosecution (and police).
Where you live, maybe -- but that's not typical for most of the world (e.g., US, UK, Canada, Australia, much of Asia including Indonesia).
In most countries (France being a notable exception) there is an clear
separation of duties between the prosecutor's office and the police. Criminal investigation & determination is up to the police and similar investigators. Only upon sufficient evidence will the police will refer charges to the prosecutor.
A prosecutor then decides whether those charges should be
prosecuted in court (hence the name). Not every crime will be prosecuted.
The prosecutor does not perform any investigative functions. If the evidence is not sufficient, the case will be referred back to police.
That is not to say that a prosecutor will simply twiddle his thumbs waiting for the police to bring charges, but it does mean a prosecutor won't normally be involved unless there is evidence of a crime.
In the AirAsia crash, there is no evidence that a crime caused the accident, and therefore no charges are pending from the National Police for a prosecutor to examine.
However:
- Since AirAsia reportedly didn't have the correct license to fly the SBY-SIN route on the day of the crash, a corruption investigation may be underway
- As the AirAsia crash involved death of a French national, a French prosecutor has opened an inquiry in Paris