Primitive ideals and nilpotent orbits in type \(G_ 2\)
Let \({\mathfrak g}\) be a finite dimensional semi-simple Lie algebra over \({\mathbb{C}}\) and J a primitive ideal of its enveloping algebra U(\({\mathfrak g})\). The associated variety \({\mathcal V}(J)\subseteq {\mathfrak g}\) * is defined as the collection of zeros of the associated graded ideal, gr J, in S(\({\mathfrak g})\), the symmetric algebra on \({\mathfrak g}\), and \({\mathcal V}(J)\) can be considered as a subvariety of \({\mathfrak g}\). Let G denote the adjoint algebraic group of \({\mathfrak g}\). If \(X\in {\mathfrak g}\) and ad X acts nilpotently on \({\mathfrak g}\), then G.X is called a nilpotent orbit. By a result of \textit{A. Joseph} [J. Algebra 93, 509-523 (1985; Zbl 0594.17009)], \({\mathcal V}(J)\) is the closure of a single nilpotent G-orbit. Now suppose \({\mathfrak g}_ 2\) is the simple Lie algebra of type \(G_ 2\). The nilpotent orbits are of dimension 0, 6, 8, 10, 12. \textit{A. Joseph} [J. Algebra 65, 284-306 (1980; Zbl 0441.17004) and Lect. Notes Math. 880, 249-266 (1981; Zbl 0468.17004)] has shown that there are precisely two completely prime primitive ideals associated with the unique nilpotent orbit of dimension 8. The paper examines this orbit and these ideals by embedding \({\mathfrak g}_ 2\) in so(7) and the Joseph ideal in U(so(7)), thus answering a number of (as yet, unpublished) questions of Vogan.
scientific article; zbMATH DE number 4049836