Quasi-reductive supergroups with small even parts

TL;DR

The paper classifies quasi‑reductive algebraic supergroups whose largest even subgroups are isomorphic to $\mathrm{GL}_2$, $\mathrm{SL}_2$, or $\mathrm{PSL}_2$, using Harish‑Chandra pairs $(G,\mathfrak{G}_{\bar{1}})$ to reduce the problem to module‑theoretic data over $G$. It provides a complete description in the $G\cong\mathrm{SL}_2$ case, identifies PSL$_2$ analogues, and gives an extensive GL$_2$‑case analysis yielding several families of deformations and parametrized families (notably ${\bf Q}(2;a,c)$, ${\bf S}(t)$, ${\bf L}(t)$, ${\bf H}(t)$, ${\bf K}(t)$), with explicit isomorphism criteria. The results are then applied to the structure of centralizers of tori, showing how these Harish‑Chandra descriptions govern the centralizers $\mathrm{Cent}_{\mathbb{G}}(T)$ and related subgroups, and suggesting that centralizers of supertori in the general quasi‑reductive setting are typically reductive. Overall, the work advances a concrete, representation‑theoretic classification framework for small‑even‑part supergroups and their torus centralizers, with several new families and deformation phenomena in characteristic various from zero.

Abstract

We describe all supergroups with the largest even supersubgroups being isomorphic to $\mathrm{GL}_2, \mathrm{SL}_2$ or $\mathrm{PSL}_2$. These results are applied to the description of centralizers of certain tori in the quasi-reductive supergroups.

Theorems & Definitions (92)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Remark 1.3
• Lemma 1.4
• proof
• Lemma 1.5
• proof
• Lemma 2.1