Journal of Lie Theory 11 (2001), No. 2, 355--379
Copyright Heldermann Verlag 2001
Subquotients in the Enveloping Algebra of a Nilpotent Lie Algebra
Bradley N. Currey III
Dept. Mathematics and Math. Computer Science, Saint Louis University, St. Louis, MO 63103, U.S.A.
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 Lf 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 Lf have dimension no greater than one.
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