Splitting quasireductive supergroups and volumes of supergrassmannians

TL;DR

This paper introduces splitting subgroups of quasireductive supergroups, provides explicit examples for certain supergroups, and computes volumes of supergrassmannians, advancing understanding of supergroup structures and their geometric properties.

Contribution

It defines splitting subgroups for quasireductive supergroups, identifies explicit examples for key cases, and computes supergrassmannian volumes, with some conjectural aspects remaining.

Findings

Explicit splitting subgroups for $GL(m|n)$, $Q(n)$, and defect one basic classical supergroups.

Proof that these subgroups are minimal up to conjugacy, except for $GL(m|n)$ where it is conjectural.

Computed volumes of complex supergrassmannians.

Abstract

We introduce the notion of splitting subgroups of quasireducitve supergroups, and explain their significance. For $GL(m|n)$, $Q(n)$, and defect one basic classical supergroups, we give explicit splitting subgroups. We further prove they are minimal up to conjugacy, except in the $GL(m|n)$ case where it remains a conjecture. A key tool in the proof is the computation of the volumes of complex supergrassmannians, which is of interest in its own right.