David M. Galban | Searcharxiv

arXiv2022

On sheaf cohomology for supergroups arising from simple classical Lie superalgebras

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 $R^{\bullet}\text{ind}_{B}^{G}L_{\mathfrak f}(λ)$ where $L_{\mathfrak f}(λ)$ 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.

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arXiv2021

On First and Second Cohomology Groups for BBW Parabolics for Classical Lie Superalgebras

Let ${\mathfrak g}$ be a classical simple Lie superalgebra. In this paper, the author studies the cohomology groups for the subalgebra $\mathfrak{n}^{+}$ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition ${\mathfrak g}={\mathfrak n}^{-}\oplus {\mathfrak f} \oplus {\mathfrak n}^{+}$ where $\mathfrak f$ is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. It is shown that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. This enables the author to explicitly compute the first and second cohomologies for ${\mathfrak n}^{+}$. The paper concludes with tables listing the weight space decompositions and dimension formulas for these cohomology groups.

math.RT↗