Dixmier Algebras On Complex Classical Nilpotent Orbits And Their Representation Theories

Abstract

For a nilpotent orbit O in a complex classical Lie group G, R. Brylinski in [7] constructed a Dixmier Algebra model of its Zariski closure, based on an earlier construction by Kraft and Procesi. On the other hand, Barbasch in [6] constructed another model on O itself. Treating G as a real Lie group with maximal compact subgroup K , both models can be seen as admissible (gC , KC )-modules of finite length. We are interested in finding out the composition factors of both models. We first list out all the possible factors that can appear in both models, and compute which of them appear in the Barbasch model. When the Zariski closure of O is normal, we prove the composition factors of the Brylinski model are the same as the Barbasch model. Also, we give a conjecture on the composition factors in the Brylinski model, irrespective of the normality of the orbit closure.