A Z2-graded generalization of Kostant's version of the Bott-Borel-Weil theorem

This paper generalizes recent work of Kostant to prove two closely related results concerning Lie superalgebras. The first result is a necessary and sufficient condition for the extension of a Lie superalgebra with non-degenerate, ad-invariant bilinear form by an orthosymplectic representation of that superalgebra. We examine when it is possible to do this, encoding the additional bracket structure using a fundamental 3-form, and determining a condition on this 3-form that is expressed in terms of the Clifford-Weyl superalgebra---the Z2-graded analog of the Clifford algebra. The second result is an extension of Kostant's algebraic form of the Bott-Borel-Weil theorem to basic classical Lie superalgebras and typical representations. The Bott-Borel-Weil theorem computes the Lie algebra cohomology of a nilpotent subalgebra as the representations of a Lie subalgebra. By careful choice of the subalgebra one may pass from accessible representations on the subalgebra to more complicated representations of the full algebra. We generalize this theorem from its Lie algebra setting to that of basic classical Lie superalgebras and the corresponding Lie superalgebra cohomology arising from these algebras with coefficients in typical highest weight representations.