On the topology of the Dixmier map
Let \(\mathfrak g\) be a finite dimensional semisimple complex Lie algebra with coadjoint group \(G\). The Dixmier map associates a primitive ideal of the universal enveloping algebra \(U({\mathfrak g})\) of \(\mathfrak g\) with any pair \((y,{\mathfrak p})\), where \(y\in{\mathfrak g}^{\ast}={\mathfrak g}\) (by the Killing form) and \(\mathfrak p\) is a polarization of \(y\) (a parabolic subalgebra). For \({\mathfrak g}={\mathfrak s}{\mathfrak l}_n\) one can forget about \(\mathfrak p\) and obtains a mapping \(\text{Dix}:{\mathfrak g}^{\ast}/G\to \mathcal X\), where the orbit space \({\mathfrak g}^{\ast}/G\) is equipped with the factor-topology of the Zariski topology and the space \({\mathcal X}\) of all primitive ideals of \(U({\mathfrak g})\) with the Jacobson topology. In this case \(\text{Dix}\) is a bijection (but not a homeomorphism) on the space \({\mathcal X}^1\) of all completely prime primitive ideals. In the paper under review the author uses a result of [\textit{W. Soergel}, Math. Z. 204, 559-581 (1990; Zbl 0685.17006)] and shows that \(\text{Dix}\) is sheet-wise a homeomorphism onto the image, where a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension (and \(\mathfrak g\) decomposes into finitely many sheets). The above result holds in the general case of any semisimple \(\mathfrak g\) when one has to restrict to a sheet of polarizable orbits where the Dixmier map can be defined. Let \(\mathbf I\) be the annihilator of the generic module induced from a fixed polarization \({\mathfrak p}\subset {\mathfrak g}\). The author describes the centre \(Z\) of the ``relative enveloping algebra'' \(U=U({\mathfrak g})/{\mathbf I}\) in terms of a relative Harish-Chandra isomorphism of the normalization \(\widetilde Z\) with a suitable ring of group invariants. He suggests that the very mild central extension \(\widetilde U\) of \(U\) by \(\widetilde Z\) enjoys good properties and is very suitable for the study of the Dixmier map. In particular, the author conjectures that for \({\mathfrak g}={\mathfrak s}{\mathfrak l}_n\) every minimal primitive ideal of \(\widetilde U\) is generated by a maximal ideal of the centre. As added in proof, this conjecture will be established in a subsequent paper.
scientific article; zbMATH DE number 1416663
scientific article