Geometry of coadjoint orbits and noncommutativity of invariant differential operators on nilpotent homogeneous spaces (Q2711833)

Let \(G\) be a connected, simply connected real Lie group with Lie algebra \({\mathfrak g}\), \(H\) a connected closed subgroup of \(G\) with Lie algebra \({\mathfrak h}\) and \(f\) a linear form on \({\mathfrak g}\) satisfying \(f([{\mathfrak h},{\mathfrak h}])= \{0\}\). Let \(\chi_f\) be the unitary character of \(H\) with differential \(\sqrt{-1}f\) at the origin and \(\tau\) the unitary representation of \(G\) induced from \(\chi_f\). The canonical central decomposition of \(\tau\) is written down with its Plancherel measure \(\mu\) in the unitary dual \(\widehat G\) of \(G\) and its multiplicity function \(m(\pi)\), \(\pi\in\widehat G\) (\(\mu\) determined almost everywhere). These ingredients are explicitly described in terms of the orbit method [cf. \textit{L. Corwin}, \textit{F. P. Greenleaf} and \textit{G. Grélaud}, Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005)]. It is well-known that the multiplicities \(m(\pi)\) are either purely infinite or uniformly bounded, accordingly we say that \(\tau\) has infinite or finite multiplicities.NEWLINENEWLINENEWLINENow, we consider the algebra \({\mathbf D}_\tau\) of \(\tau(G)\)-invariant differential operators on the line bundle with base space \(G/H\) associated to these data. In their fundamental paper [Comm. Pure Appl. Math. 45, 681-748 (1992; Zbl 0812.43004)] \textit{L. Corwin} and \textit{F. P. Greenleaf} proved that \({\mathbf D}_\tau\) is commutative if \(\tau\) has finite multiplicities, and conjectured that the converse is also true. This commutativity conjecture is an interpretation for the nilpotent case of \textit{M. Duflo's} conjecture [Open problems in the representation theory of Lie groups, Conference on ``Analysis on homogeneous spaces'' (Katata/Japan), T. Oshima (ed.), 1-5 (1986)] formulated for a more general setting.NEWLINENEWLINENEWLINEIn this paper, the author proves the conjecture under some additional assumptions. His methods rest on the analysis of coadjoint orbit geometry. It is on the line of his research that we have finally established the commutativity conjecture in the nilpotent case [cf. \textit{H. Fujiwara}, \textit{G. Lion}, \textit{B. Magneron} and \textit{S. Mehdi}, Math. Ann. 327, 513-544 (2003)].