crucial role in the representation theory of $B$ and $N$ (see, e.g., [@Kirillov1], [@Kirillov2]). Works of J. Dixmier, M. Duflo, M. Vergne, O. Mathieu, N. Conze and R. Rentschler led to the result that the orbit method provides a nice description of primitive ideals of the universal enveloping algebra of a nilpotent Lie algebra (in particular, of ${\mathfrak{n}}$). Let us describe this in detail.

Let ${\mathfrak{n}}^*$ be the dual space of ${\mathfrak{n}}$. To any linear form $\lambda\in{\mathfrak{n}}^*$ one can assign a bilinear form $\beta_{\lambda}$ on ${\mathfrak{n}}$ by putting $\beta_{\lambda}(x,y)=\lambda([x,y])$. A subalgebra ${\mathfrak{p}}\subseteq{\mathfrak{n}}$ is a *polarization of ${\mathfrak{n}}$ at* $\lambda$ if it is a maximal $\beta_{\lambda}$-isotropic subspace. By [@Vergne], such a subalgebra always exists. Let ${\mathfrak{p}}$ be a polarization of ${\mathfrak{n}}$ at $\lambda$, and $W$ be the one-dimensional representation of ${\mathfrak{p}}$ defined by $x\mapsto\lambda(x)$. Then the annihilator $J(\lambda)={\mathrm{Ann}\,}_{U({\mathfrak{n}})}{V}$ of the induced representation $V = U({\mathfrak{n}}) \otimes_{U({\mathfrak{p}})} W$ is a primitive two-sided ideal of $U({\mathfrak{n}})$. It turns out that $J(\lambda)$ depends only on $\lambda$ and not on the choice of polarization. Further, $J(\lambda)=J(\mu)$ if and only if the coadjoint $N$-orbits of $\lambda$ and $\mu$ coincide. Finally, the *Dixmier map* $${\mathcal{D}}\colon{\mathfrak{n}}^*\to{\mathrm{Prim}\,}U({\mathfrak{n}}),~\lambda\mapsto J(\lambda),$$ induces a homeomorphism between ${\mathfrak{n}}^*/N$ and ${\mathrm{Prim}\,}U({\mathfrak{n}})$, where the latter set 