Infinite Kostant cascades and centrally generated primitive ideals of $U(\mathfrak{n})$ in types $A_{\infty}$, $C_{\infty}$

Abstract: We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{sl}{\infty}(\mathbb{C})$, $\mathfrak{so}{\infty}(\mathbb{C})$, $\mathfrak{sp}{\infty}(\mathbb{C})$. There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we provide explicit generators of the center of $U(\mathfrak{n})$ in all cases. We then fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$ and characterize the centrally generated primitive ideals of $U(\mathfrak{n})$ for $\mathfrak{g}=\mathfrak{sl}{\infty}(\mathbb{C})$, $\mathfrak{sp}{\infty}(\mathbb{C})$ 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 $\mathfrak{sl}_n(\mathbb{C})$, $\mathfrak{sp}{2n}(\mathbb{C})$.