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The main focus of this paper is Bott-Borel-Weil (BBW) theory for basicclassical Lie superalgebras. We take a purely algebraic self-contained approachto the problem. A new element in this study is twisting functors, which we usein particular to prove that the top of the cohomology groups of BBW theory forgeneric weights is described by the recently introduced star action. We alsostudy the algebra of regular functions, related to BBW theory. Then weintroduce a weaker form of genericness, relative to the Borel subalgebra andshow that the virtual BGG reciprocity of Gruson and Serganova becomes an actualreciprocity in the relatively generic region. We also obtain a completesolution of BBW theory for osp(m|2), D(2,1;a), F(4) and G(3) with distinguishedBorel subalgebra. Furthermore, we derive information about the category offinite dimensional osp(m|2)-modules, such as BGG-type resolutions and Kostanthomology of Kac modules and the structure of projective modules.

Bott-Borel-Weil theory, BGG reciprocity and twisting functors for Lie superalgebras