On the enright functor in the highest weight category

Abstract

LetG be a complex semisimple Lie group,B its Borel subgroup andX a flag variety ofG. We define a functor on the category ofB-equivariantD X-modules that corresponds, under the global section functor, to the Enright functor on the highest weight category. We use this to lift Enright functor to the mixed version of the highest weight category. As an application we obtain that the socle and the cosocle filtration of a primitive quotient of the enveloping algebra coincide.