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

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 $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathfrak{s}\mathfrak{o}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathfrak{s}\mathfrak{p}}_{\mathrm{\infty }}(\mathbb{C})$. 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 $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathfrak{s}\mathfrak{p}}_{\mathrm{\infty }}(\mathbb{C})$ 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 ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{C})$, ${\mathfrak{s}\mathfrak{p}}_{2n}(\mathbb{C})$.