The ASA or Air Source Adapter is the part you screw your air tank into.
The company that make the Invert Mini or Empire Mini as it now known as are called KEE Action Sports, they also make a marker similar to the Mini called the Empire Axe.
The ASA on an Axe is similar to the Mini, doubling as the marker's operating pressure regulator. But it has a built in on/off lever which allows you to purge the excess air in the marker, making it easier to remove the bottle when you're done playing.
It makes a nice upgrade to the Mini, it's called the Empire Relay On/Off ASA. Look it up on the traders above the thread content, Just Paintball, LiPS and so on will usually have them in stock.
The saying "If it ain't broke, don't fix it" while true, there are certain things you can do to improve the performance.
Lightweight bolts for example, enable you to lower the operating pressure of the Mini. On my Axe, the Lurker Eigenbolt I have lets me tweak the operating pressure down from 200psi to 130psi. It means a quiet and efficient gat with little to no kick. It's awesome
Have a google on it, read up what various people have done. Weigh up the pros and cons and if its what you feel like doing, why not go for it ?