The lawsuit -- to recover the costs of investigating the case -- outlines what the City claims Jussie did. It says Jussie and Abel Osundairo had been friends since 2017 -- socializing and exercising together. It says Abel assisted Jussie in obtaining recreational drugs.

The lawsuit claims Jussie told Abel shortly before the "attack," "Might need your help on the low."

The lawsuit claims, as we reported, Jussie scoped out the location for the "attack" with Abel and Ola 2 days before the attack and a dry run the day before. They agreed Abel would attack Jussie but would not hurt him too badly and would give Jussie a chance to appear to fight back.

The suit claims they also scripted the entire attack ... starting with the brothers calling Jussie "Empire f***** Empire n*****" -- referring to the homophobic and racist slurs.

They discussed the brothers placing a rope around Jussie's neck and pouring a liquid on him.

The brothers were also instructed not to bring their cellphones -- presumably, so they couldn't be tracked.

The suit says Jussie intentionally misled officers into believing his attackers were white, when he knew otherwise.

As we reported ... the city threatened to sue Jussie if he didn't fork over $130k to cover the cost of investigating a case it says was faked by the "Empire" star. The City of Chicago fired off a letter to Jussie's people earlier this month ... but his camp told the City to go pound sand.

The City says the cost of investigating the case was at least $130,000 in overtime alone, but it wants hundreds of thousands of dollars on top of that to cover attorneys fees and additional expenses. And then the City wants the judge to multiply that times 3, to punish Jussie.