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Journal of Lie Theory, Vol. 11, No. 2, pp. 355-379 (2001)
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#
Subquotients in the Enveloping Algebra of a Nilpotent Lie Algebra

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Bradley N. Currey III

Department of Mathematics and Mathematical Computer Science

Saint Louis University

St. Louis, MO 63103

U.S.A.

curreybn@slu.edu

**Abstract:** For any triple $(\frak g, \frak h, f)$ where $\frak g$ is a nilpotent Lie algebra over a field $\bold k$ of characteristic zero, $\frak h$ is a subalgebra of $\frak g$, and $f$ is a homomorphism of $\frak u(\frak h)$ onto $\bold k$, a subquotient ${\cal D}(\frak g, \frak h, f)$ of $\frak u(\frak g)$ 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 $\exp(\ad^*\frak h)$-orbits in the variety $L_f$ of linear functionals in $\frak g^*$ whose restriction to $\frak h$ agree with $f$. Certain constructions lead to a procedure by which the question of non-commutativity of ${\cal D}(\frak g, \frak h, f)$ is reduced to a case where $(\frak g, \frak h, f)$ has a special structure. This reduction is then used to prove that the Corwin-Greenleaf conjecture about non-commutativity of ${\cal D}(\frak g, \frak h, f)$ holds in certain situations, in particular when the $\exp(\ad^*\frak h)$-orbits in $L_f$ have dimension no greater than one.

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