Let \(G\) be a Lie group. Let \(H\) be a Lie subgroup of \(G\) with Lie algebra \(\mathfrak{h}\). Let \(\mathfrak{h}^*\) be the vector space dual of \(\mathfrak{h}\). Let \(\pi\) be a unitary irreducible representation of \(G\). According to the philosophy of the orbit method, there should correspond to \(\pi\) a coadjoint orbit \(\mathcal{O}_{\pi}\) such that the \(H\)-orbits occurring in the image of the associated moment map \(\operatorname{p}:\mathcal{O}_{\pi}\rightarrow\mathfrak{h}^*\) determine the spectrum of the restriction \(\pi\arrowvert_ H\).
For discrete series representations of almost algebraic real Lie groups, a refinement of this principle is the Duflo conjecture, which gives in terms of a properness condition on the moment map, a criterion for \(\pi\arrowvert_H\) to admit a direct sum decomposition into unitary irreducible representations of \(H\), each having a finite multiplicity itself determined by the geometry of the reduced space of \(\mathcal{O}_{\pi}\) relative to p.
When \(G\) is a general linear group over the real or complex numbers and \(H = P\) is its mirabolic subgroup, the Kirillov conjecture, proved in full generality by E. M. Baruch [Ann. Math. (2) 158, No. 1, 207–252 (2003; Zbl 1034.22010)], states that the restriction \(\pi\arrowvert_P\) is irreducible. In this setting, the results of the paper under review are in the spirit of the aforementioned ideas: it is shown that \(\pi\arrowvert_P\) is exactly the unitary irreducible representation which corresponds in Duflo’s orbit method to the unique Zariski open and dense \(P\)-orbit \(\Omega\) in \(\operatorname{p}(\mathcal{O}_{\pi})\). This provides a geometric interpretation of the Kirillov conjecture. It is also shown that the moment map \(\operatorname{p}\) is proper over \(\Omega\) and the reduced space of \(\Omega\) is a singleton set. As \(G\) does not admit discrete series representations, this verifies a generalisation of the Duflo conjecture, supporting in particular the conjectural relationship between multiplicities and the geometry of the reduced space.
The results of the paper depend on a detailed analysis of the geometry of the moment map, which is evaluated using the classification of coadjoint orbits of \(P\) obtained in [G. Liu and J. Yu, “A Geometric Interpretation of Kirillov’s Conjecture”, Preprint, arXiv:1806.06318], where similar results are obtained for tempered representations of \(G\).