Rather than use an inverter to power the air conditioner, you could take an air conditioner unit that is readily designed to run from DC voltage available from solar power systems.
Many companies are making DC powered air conditioner units, because for the manufacturer, it's a relatively small design change to the unit. Basically all AC units today are 'invergter driven' e.g. they have a frequency changer to drive the motor(s). That already runs from a DC voltage, which is taken from a rectifier stage, and just needs to be tweaked to accept DC input at a suitable voltage.
Here's one such solution from one manufacturer (GE innovations) that I found from Google:
Using a DC powered air conditioner unit eliminates redundant power conversions making it more efficient. It's also much more easier since the air conditioner unit can do decisions such as low voltage cut off much easier than a separate inverter which must work on a very large operating range.
Finally, solar power and cooling is a perfect match because you need the most cooling power exactly when the sun is shining and solar power is available! When it's cloudy and the solar cells generate less power, you also need less cooling!