Splitting quasireductive supergroups and volumes of 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.