Authors’ summary: ”The characters (extremal positive definite central functions) of (infinite) discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group \(C^*(G)\) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1-1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.”