Basic quasi-reductive root data and supergroups

Abstract

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is isomorphic to $G$, and the quotient $\mathbb{G}/\mathbb{G}_{\text{ev}}$ is $G$-equivariantly isomorphic to $Y$. We prove that, if $Y$ satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain $0$; (ii) $\mathfrak{g}:=\text{Lie}(\mathbb{G})$ admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type.