Representation theory on the open Bruhat cell

The action of a reductive algebraic group G on G/P−, where P− is a parabolic subgroup, differentiates to a representation of the Lie algebra g of G by vector fields on U+, the unipotent radical of a parabolic opposite to P−. We show that both the regular functions on U+ and the polynomial vector fields on U+ form g-modules that are dual to parabolically induced modules, construct an explicit composition chain of the former module in the case where G is classical simple and U+ is Abelian, and indicate how this chain can be used to analyse the the latter module.