Subquotients in the enveloping algebra of a nilpotent Lie algebra (Q2730487)

Let \(G\) be a connected simply connected nilpotent real Lie group, \(H\) an analytic subgroup of \(G\) and \(\chi\) a unitary character of \(H\). Let \(\tau= \text{Ind}_H^G\chi\). Then the conjecture of Duflo-Corwin-Greenleaf states that the algebra \(D_\tau(G/H)\) of \(C^{\infty}\) \(G\)-invariant differential operators on the space \(G/H\) is commutative if and only if the multiplicities occurring in the decomposition into irreducibles of \(\tau\) are finite. This conjecture was the subject of many works and has been recently solved. In the paper under review, the author defines a generalized version of the above conjecture.NEWLINENEWLINENEWLINEMore precisely, let \(\mathfrak g\) be a nilpotent Lie algebra over a field \(\mathbf k\) of characteristic zero, \(\mathfrak h\) a subalgebra of \(\mathfrak g\) and \(f\) a homomorphism of \({\mathcal U}(\mathfrak h)\) onto \(\mathbf k\). So a subquotient \({\mathcal D}(\mathfrak g, \mathfrak h,f)\) of the universal enveloping algebra of \(\mathfrak g\) is constructed and considered to be a generalization of the algebra of invariant differential operators \(D_\tau(G/H)\) mentioned above. The author also defines the notion of an IM triple \((\mathfrak g, \mathfrak h,f)\) (infinite multiplicity case) and equivalently the notion of an FM triple \((\mathfrak g, \mathfrak h,f)\) (finite multiplicity case). The generalized conjecture consists in stating that for an IM triple \((\mathfrak g, \mathfrak h,f)\), the algebra \({\mathcal D}(\mathfrak g, \mathfrak h,f)\) is not commutative. Under suitable conditions, the conjecture is proved. In the proofs, the author reduces the original triple \((\mathfrak g, \mathfrak h,f)\) to another triple \((\mathfrak g', \mathfrak h',f')\) having better structure and for which all the necessary information about orbit dimensions and \({\mathcal D}(\mathfrak g, \mathfrak h,f)\) is preserved.