Lowest court in NY?!
This is the State Supreme Court.
Confusing, isn't it? Not being from New York, I can't always keep them straight, but I think it works like this:
In New York "Supreme Court" is the trial-level court of general jurisdiction in the state court system. It is one of several courts that are "Superior Courts" The first level of appeal from Superior (Supreme) court is the Supreme Court, Appellate Terms and above that is the Supreme Court, Appellate Division. At the top, the New York State Court of Appeals.
(At this point, I'm not sure which court is hearing these cases.)