On some representations of nilpotent Lie algebras and superalgebras

TL;DR

This paper explores the representation theory of nilpotent Lie algebras and superalgebras, linking geometric properties of coadjoint orbits to module counts and analyzing algebraic structures of primitive ideals.

Contribution

It determines the number of modules associated with two-dimensional coadjoint orbits and describes the structure of factor algebras of universal enveloping superalgebras.

Findings

Exact count of modules for two-dimensional orbits

Identification of cases with purely Weyl algebra factors

Dependence of Weyl algebra sizes on primitive ideals

Abstract

Let $G$ be a simply connected, nilpotent Lie group with Lie algebra $\gee$. The group $G$ acts on the dual space $\gee^*$ by the coadjoint action. %% which partitions $\gee^*$ into coadjoint orbits. By the orbit method of Kirillov, the simple unitary representations of $G$ are in bijective correspondence with the coadjoint orbits in $\gee^*$, which in turn are in bijective correspondence with the primitive ideals of the universal enveloping algebra of $\gee$. The number of simple $\gee$-modules which have a common eigenvector for a particular subalgebra of $\gee$ and are annihilated by a particular primitive ideal $I$ is shown by Benoist to depend on geometric properties of a certain subvariety of the coadjoint orbit corresponding to $I$. We determine the exact number of such modules when the coadjoint orbit is two-dimensional. Bell and Musson showed that the algebras obtained by factoring the universal enveloping superalgebra of a Lie superalgebra by graded-primitive ideals are isomorphic to tensor products of Weyl algebras and Clifford algebras. We describe certain cases where the factors are purely Weyl algebras and determine how the sizes of these Weyl algebras depend on the graded-primitive ideals.