The Lambert Ring territories are a world that were presumably part of the Outback or protectorate during the Web Era. Based on the few descriptions in the Cantos books, the territories presumably consist of clusters of mining communities living in zero or low-gravity terraformed asteroids, possibly in a ring around a planet in the likes of Saturn on Sol System. One of these asteroids was called Peregrine, that was part of the Peregrine Trojan Cluster. The primary economic activity in the Lambert Ring territories seems to have bee asteroid mining and trading.
At least some of the population in the Ring belonged to a group called the Free Miners. In his early military days, Colonel Fedmahn Kassad (then Lieutenant Kassad) assisted in the evacuation of Hegemony consulate staff and citizens during an undeclared war with the Free Miners. He did so but cutting through the bottom of the “old asteroid bore shaft” on Peregrine. Given the presence of the Hegemony consulate on Peregrine, perhaps that asteroid is the capital of the Lambert Ring territories.
SPOILER ALERT! The section(s) below are MAJOR SPOILERS.
During the Pax Era, the Lambert Ring territories were a predominantly non-Christian world.  The clusters were stage to a rebellion by the free traders against the Pax as a result of the increasing resistance against the Catholic Church in the events leading up to the Shared Moment.
As non-Christians, Lambert's entire population was at one point subject to being "rendered lifeless" and transported elsewhere in a joint action by the Pax and the TechnoCore, along with the inhabitants of several other worlds. 
Rhadamanth Nemes, when speaking with other Pax officials, claimed that most of her military training was done on Lee Three and on the Peregrine Trojan Cluster in the Lambert Ring territories. In the Ring region, soldiers trained in zero-g or low-g environments. Accordingly, it seems that the Lambert worlds were of military importance to the Pax, perhaps as military training base.