Infinite Kostant cascades and centrally generated primitive ideals of U(n)U(n) in types A∞A∞, C∞C∞

We study the center of U(n)U(n), where nn is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra g=sl∞(C)g=sl∞(C), so∞(C)so∞(C), sp∞(C)sp∞(C). There are infinitely many isomorphism classes of Lie algebras nn, and we provide explicit generators of the center of U(n)U(n) in all cases. We then fix nn with “largest possible” center of U(n)U(n) and characterize the centrally generated primitive ideals of U(n)U(n) for g=sl∞(C)g=sl∞(C), sp∞(C)sp∞(C) in terms of the above generators. As a preliminary result, we provide a characterization of the centrally generated primitive ideals in the enveloping algebra of the nilradical of a Borel subalgebra of sln(C)sln(C), sp2n(C)sp2n(C).