On sheaf cohomology for supergroups arising from simple classical Lie superalgebras

Abstract

In this paper the authors study the behavior of the sheaf cohomology functors $R^{\bullet }{\text{ind}}_B^G(-)$ where $G$ is an algebraic group scheme corresponding to a simple classical Lie superalgebra and $B$ is a BBW parabolic subgroup as defined by D. Grantcharov, N. Grantcharov, Nakano and Wu. We provide a systematic treatment that allows us to study the behavior of these cohomology groups $R^{\bullet }{\text{ind}}_B^G{L}_{\mathfrak{f}}(\lambda )$ where $L_{\mathfrak{f}}(\lambda )$ is an irreducible representation for the detecting subalgebra $\mathfrak{f}$. In particular, we prove an analog of Kempf's vanishing theorem and the Bott-Borel-Weil theorem for large weights.