Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras. For each choice of parabolic subalgebra and representation of such a Lie superalgebra, there is a natural definition of the boundary and coboundary operators, which define (co)homology of the nilradical of the parabolic subalgebra. We prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of (generalised) Verma modules. Every such a resolution is then given by modules induced by these homology groups. We also prove that if these homology groups are completely reducible, a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the Kostant quabla operator, which is equivalent with disjointness of the boundary and coboundary operators. Then we use these results to derive very explicit conditions 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. We also apply the obtained theory to construct specific examples of BGG resolutions for osp(m|2n). Finally we state some results for typical modules of gl(m|n) and osp(2|2n).