Quantization of Hypersurface Orbital Varieties in sl(n)

Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest weight representations of g. It is known that all unitary highest weight representations can be obtained by their quantization [8]. A hypersurface orbital variety is one which is of codimension 1 in the nilradical of a parabolic. Their classification for g = sl(n) is obtained from general results in [12]. Recently Benlolo and Sanderson [2] conjectured the form of the (nonlinear) element describing such a variety. This paper proves that conjecture and further constructs a simple module with integral highest weight which “quantizes” the variety in the precise sense that the regular functions of its closure is given a g module structure compatible (up to shift by the highest weight) with its h module structure.