Subquotients in the Enveloping Algebra of a Nilpotent Lie Algebra

B. N. Currey III

Journal of Lie theory, Tome 11 (2001) no. 2, pp. 355-379.

Résumé

For any triple (g, h, f) where g is a nilpotent Lie algebra over a field k of characteristic zero, h is a subalgebra of g, and f is a homomorphism of u(h) onto k, a subquotient D(g, h, f) of u(g) is studied which generalizes the algebra of invariant differential operators on a nilpotent homogeneous space. A generalized version of a conjecture of Corwin and Greenleaf is formulated using geometry of exp( ad* h)-orbits in the variety L_f of linear functionals in g* whose restriction to h agree with f. Certain constructions lead to a procedure by which the question of non-commutativity of D(g, h, f) is reduced to a case where (g, h, f) has a special structure. This reduction is then used to prove that the Corwin-Greenleaf conjecture about non-commutativity of D(g, h, f) holds in certain situations, in particular when the exp(ad* h)-orbits in L_f have dimension no greater than one.