Cohomology and Representation Theory for Lie Superalgebras

In this paper the authors describe two general frameworks for studying algebraic supergroup and Lie superalgebra representations. These encompass the well-known rational representations for a reductive algebraic group and the parabolic Category $\mathcal{O}$ for semisimple Lie algebras. Many of the important developments will be surveyed and a blueprint for these new approaches will be provided. The existence of odd elements simply add an additional layer of difficulty. The authors prove that the calculation of the sheaf cohomology in the super setting for the rational category is equivalent to understanding the Lie superalgebra cohomology. This generalizes the well-known fact that the classical Bott-Borel-Weil theorem is equivalent to Kostant’s theorem. Furthermore, for the parabolic Category ${\mathcal{O}}^{\mathfrak{p}}$ where $\mathfrak{p}$ is a principal parabolic, the authors show that the characters of simple modules can be obtained by looking at natural generalizations of Kazhdan-Lusztig polynomials given by extensions between simple modules and costandard modules arising from the underlying costandardly stratified category.