Generic representations of classical Lie superalgebras and their localization

Abstract

We complete the study of arbitrary generic irreducible modules over a classical complex Lie superalgebraG initiated in [14] (whereG was assumed to be of type I) by presenting a full description of the underlyingG0-module of any suchG-module. This enables us in particular to extend Beilinson-Bernstein's localization theorem to a certain full subcategory of the category ofG-modules with fixed central character and also to describe the image of the enveloping algebraU(G) in the global sections of a generic twisted ring of differential operators on any flag superspace. As an application we construct an infinite family of full subcategories of the category ofG-modules with fixed generic atypical central character, each of which is equivalent to the category ofG0-modules with fixed regular central character.