0&-e_{{\epsilon}_n}&\ldots&-e_{{\epsilon}_2}&-e_{{\epsilon}_1}\\
\end{vmatrix}$$ (our normalization is such that the term $e_{{\epsilon}_n}\xi_{{\epsilon}_{n-2}+{\epsilon}_{n-1}}$ enters $\xi_{{\epsilon}_n}$ with coefficient 1).

For $A_{n-1}$ and $C_n$, we denote $\Delta_{\beta}=\sigma(\xi_{\beta})$ for $\beta\in{\mathcal{B}}$. Since all $e_{\alpha}$ involved in $\xi_{\beta}$ (i.e., $e_{\alpha}$ which appear in a term of $\xi_{\beta}$) commute, we conclude that $\Delta_{\beta}$, $\beta\in{\mathcal{B}}$, is defined as an element of $U({\mathfrak{n}})$ again by formulas (\[formula:Delta\_i\_A\_n\]), (\[formula:Delta\_i\_C\_n\]) for $A_{n-1}$, $C_n$ respectively. For $B_n$ and $D_n$, we denote $P_{\beta}=\sigma(\xi_\beta)$ for $\beta\in{\mathcal{B}}$. If $i$ is odd, all $e_{\alpha}$ involved in $\xi_{{\epsilon}_i+{\epsilon}_{i+1}}$ commute, so the polynomial $P_{{\epsilon}_i+{\epsilon}_{i+1}}$ is defined as an element of $U({\mathfrak{n}})$ again by formula (\[formula:Delta\_i\_D\_n\_pf\]).

**Definition ..** [For $A_{n-1}$ and $C_n$ (respectively, for $B_n$ and $D_n$), we call $\Delta_{\beta}$ (respectively, $P_{\beta}$), $\beta\in{\mathcal{B}}$, the *canonical generators* of $Z({\mathfrak{n}})$.]{}

**.. [Centrally generated ideals of $U({\mathfrak{n}})$]{}.**

Throughout\[sst:ideals\_fd\] this subsection ${\mathfrak{g}}$ and ${\mathfrak{n}}$ are as in Subsection \[sst:center\_fd\]. By definition, an ideal $J\subseteq U({\mathfrak{n}})$ is *primitive* if $J$ is the annihilator of a simple ${\mathfrak{n}}$-module. Here we describe all primitive *centrally generated* ideals of $U({\mathfrak{n}})$, i.e., all primitive ideals $J$ generated (as ideals) by their intersections $J\cap Z({\mathfrak{n}})$ with the center $Z({\mathfrak{n}})$ of $U({\mathfrak{n}})$.

In the 1960s A. Kirillov, B. Kostant and J.-M. Souriau discovered that the orbits of the coadjoint action play a 