Zur Topologie der Dixmier-Abbildung

Summary

LetG be the coadjoint group of a finite-dimensional complex Lie algebrag. Forg solvable, the Dixmier-map is known to be a homeomorphism$\mathfrak{g}*/G \to \chi$ of the orbit space$\mathfrak{g}*/G$ /G onto the space χ of primitive ideals in the enveloping algebra U(G) [6,15]. For$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n$, the Dixmier-map is known to be a bijection (and in general not a homeomorphism)$\mathfrak{g}*/G \to \chi ^l$ with the space χ^l of all completely prime primitive ideals [7, 16]. Here we derive from a result ofW. SOERGEL [18], that this map issheet- wise a homeomorphism onto the image. Here a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension; obviouslyg decomposes into finitely many sheets [3]. The results of this paper hold more generally forg semisimple, if one restricts to a sheet of polarizable orbits, where a Dixmier-map can be defined.

Relative to a fixed polarization (a parabolic subalgebra)p ⊂g let I be the annihilator of the generic module induced from p. The „relative enveloping algebra“$U: = U(\mathfrak{g})/I$) has been studied e.g. bySOERGEL [19, 18]. Its center Z is described here by a relative Harish-Chandra isomorphism of the normalization$\tilde Z$ with a suitable ring of group invariants (3.2). We study here the extension$\tilde U$ ofU by$\tilde Z$. We suggest that this very mild central extension ofU generates good properties and is very suitable for the study of the Dixmier-map (cf.4.3,5.6).

In particular, we conjecture in case$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n$: Every minimal primitive ideal of$\tilde U$ is generated by a maximal ideal of the center. This would generalize for$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n$ a well known theorem ofM. Duflo (casep Borel, where$\tilde U = U = U(\mathfrak{g})$).

AsJ. Dixmier communicated in a letter, the main result here is exactly what he had hoped for when he first introduced a notion of sheets many years ago.