Mathematics – Representation Theory
Scientific paper
2003-11-26
The orbit method in geometry and physics (Marseille, 2000), 165--196, Progr. Math., 213, Birkhauser Boston, Boston, MA, 2003
Mathematics
Representation Theory
36 pages
Scientific paper
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 in this fashion. A hypersurface orbital variety is one which is of codimension 1 in the nilradical of a parabolic. Their classification for g = sl(n) obtains from general results. Recently Benlolo and Sanderson conjectured the form of the (non-linear) 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.
Joseph Anthony
Melnikov Anna
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