The minimal realization from deformation theory
Let \(\mathfrak g\) be a complex simple Lie algebra not of type \(A_n\). Then \(\mathfrak g\) has a unique coadjoint orbit \(\mathcal O\) of minimal nonzero dimension and the enveloping algebra \(U(\mathfrak g)\) of \(\mathfrak g\) has a unique completely prime primitive ideal \(J\) whose associated variety is the Zariski closure \(\overline{\mathcal O}\) of \(\mathcal O\) [Ann. Sci. Éc. Norm. Supér. (4) 9, 1-29 (1976; Zbl 0346.17008)]. The purpose of this paper is to give a realization of \(J\) via Gerstenhaber deformation of the coordinate ring \(A:=S(\mathfrak g)/I\) of the variety \(\overline{\mathcal O}\). This is greatly simplified by the fact that \(A\) is a Koszul algebra, as shown by Bezrukavnikov and Imandar-Mehta. At the same time, the author is able to construct modules for the deformed algebra \(U(\mathfrak g)/J\) and in particular to realize \(J\) in a uniform way as the annihilator of a simple highest weight module. He also recovers an old result of Garfinkle that the graded ideal \(\text{gr }J\) in \(\text{gr }U(\mathfrak g)=S(\mathfrak g)\) coincides with \(I\), using a result of Kostant that \(I\) is generated by quadratic polynomials.
scientific article; zbMATH DE number 1218704
scientific article