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).Comment: 36 pages. arXiv admin note: text overlap with arXiv:1209.621
