Notes on the Duflo-Serganova functor in positive characteristic
A. N. Zubkov
TL;DR
This work extends the Duflo-Serganova functor to fields of odd positive characteristic, building a framework to compute the symmetry supergroup $\widetilde{\mathbb{G}_x}$ for a broad class of supergroups and square-zero odd elements $x$. It develops a de Rham/Cartier-type approach to compute the coordinate Hopf superalgebra of $\widetilde{\mathbb{G}_x}$ and provides explicit structural descriptions in key cases such as $\mathrm{GL}(m|n)$ and $\mathrm{Q}(n)$, including minimal and maximal rank of $x$; it also analyzes the injection of $\mathbb{G}_x$ into $\widetilde{\mathbb{G}_x}$ and constructs faithful finite-dimensional representations capturing the extended symmetry. The paper extends the DS-functor to superschemes, proves fundamental functorial properties, and derives crucial auxiliary results on $\mathrm{Sym}(V)_x$ and Cartier isomorphisms, enabling explicit computations in positive characteristic. Furthermore, it investigates when a $\mathbb{G}$-supermodule is injective via the DS-functor, establishing a criterion in quasi-reductive settings with distinguished parabolics, and situates these results within a higher-weight category framework for finite-dimensional modules. Overall, the results illuminate how positive characteristic enhances DS-functor symmetry, influence injectivity criteria, and shape the associated representation theory of classical supergroups.
Abstract
We develop a fragment of the theory of Duflo-Serganova functor over a field of odd characteristic. We elaborate a method of computing the symmetry supergroup $\widetilde{\mathbb{G}_x}$ of this functor, recently introduced by A.Sherman, for a wide class of supergroups $\mathbb{G}$, and apply it to the case when $\mathbb{G}$ is $\mathrm{GL}(m|n)$ or $\mathrm{Q}(n)$, and a square zero odd element $x\in \mathrm{Lie}(\mathbb{G})$ has minimal or maximal rank. For any quasi-reductive supergroup $\mathbb{G}$, which has a pair of specific parabolic supersubgroups, we prove a criterion of injectivity of a $\mathbb{G}$-supermodule, involving vanishing of Duflo-Serganova functor on it.