Orbital varieties of the minimal orbit

Abstract

Let g be a complex simple Lie algebra with triangular decomposition g = n + ⊕ h ⊕ n −. For any nilpotent orbit O an orbital variety V of O is defined to be an irreducible component of n + ∩ O. We say that V is strongly (resp. weakly) quantizable if there exists a U( g ) module L isomorphic to R[ V ] as a U( h ) module, up to a weight shift (resp. whose associated variety is V ). Here we obtain an explicit necessary and sufficient condition for strong (resp. weak) quantization of an orbital variety of the minimal non-zero nilpotent orbit. This shows that there is always at least one orbital variety admitting strong quantization, a result which hopefully should hold for any nilpotent orbit as the corresponding annihilator would be completely prime. On the other hand it also shows that even weak quantization can fail and even when this holds strong quantization can fail. In this latter case using the Demazure operators we show exactly how close the formal character of R[ V ] can approach that of a U( g ) module and suggest that a similar behaviour holds in general.