Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

Abstract

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resolution is then given by modules induced by these cohomology groups. We also prove that if these cohomology groups are completely reducible, a sufficient condition for the existence of such a resolution is that these groups are isomorphic to the kernel of the Kostant quabla operator, which is equivalent with disjointness of the derivative and coderivative. Then we use these results to derive very explicit criteria under which BGG resolutions exist, which are particularly useful for the superalgebras of type I. For the unitarisable representations of gl(m|n) and osp(2|2n) we derive conditions on the parabolic subalgebra under which the BGG resolutions exist. This extends the BGG resolutions for gl(m|n) previously obtained through superduality and leads to entirely new results for osp(2|2n). We also apply the obtained theory to construct specific examples of BGG resolutions for osp(m|2n).