Most people will probably think heavy gear oil (aka diff oil gl-5 spec) is the go, but I'd advise against it. It will work, but it won't be ideal.

Either use a monograde engine oil, or a gl4 spec gearbox oil.

If price isn't a big deal, about the best gearbox oil (and will also be good for this app) is shell spirax S4 txm

The 'short' version is diff oil (gl 5 spec oil) has an additive package to make it 'work' with diff gears that are hypoid (the pinion rotates on a different centreline to the crownwheel, so the teeth don't 'mesh' like gearbox teeth, they actually slide across each other) - this requires certain additives to make it work which are actually not as good for 'regular' gears and also corrosive to certain white/yellow metals (of which there might be some inside the front case of the supercharger or there might not).

Diff oil is also thicker at room temp, but actually thins out a LOT at operating temp.It's also rated on a different viscosity scale/range than regular oil, so in fact 80w/90 diff oil thins to similar to 5w-30 engine oil at 100C.

If monograde engine oil or gl-4 spec gearbox oil isn't easy or cheap to get where you are, I'd suggest dexron iii automatic transmission fluid (heck I'd suggest dex ii but I don't know if it's even available anymore. The current spec last time I checked is dexron vi - which is apparently backwards compatble with autos requiring dex ii or iii, but it is a little bit thinner. )

Another thing I'd add is that just as important as 'which oil' (maybe even more so) is to regularly make sure it's got enough oil in there, and if you can, change it every time you change the engine oil, or every 5000km if you aren't changing the engine oil as often. Since they only take a few hundred ml of oil (approximately, probably a bit less) it's not that expensive to change it that often.

I'd add that I'd probably change the oil on an amr500 more often than an sc12 or sc14 charger - because they are smaller they have to be spun at higher rpm to make enough boost, so they will tend to see a little more heat potentially.