On sheaf cohomology for supergroups arising from simple classical Lie superalgebras

TL;DR

This paper investigates sheaf cohomology functors for algebraic supergroups linked to simple classical Lie superalgebras, establishing analogs of key theorems like Kempf's vanishing and Bott-Borel-Weil.

Contribution

It provides a systematic framework for studying cohomology groups of irreducible representations in the supergroup setting, extending classical theorems to this context.

Findings

Proved an analog of Kempf's vanishing theorem for supergroups.

Established a Bott-Borel-Weil theorem for large weights.

Analyzed the behavior of sheaf cohomology functors for algebraic supergroups.

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.