Invariant Berezin Integration on Homogeneous Supermanifolds

Let G be a Lie supergroup and H a closed subsupergroup.We study the unimodularity of the homogeneous supermanifold G/H , i.e. the existence of G -invariant sections of its Berezinian line bundle.To that end, we express this line bundle as a G -equivariant associated bundle of the principal H -bundle G G/H .We also study the fibre integration of Berezinians on oriented fibre bundles.As an application, we prove a formula of 'Fubini' type:. Moreover, we derive analogues of integral formulae for the transformation under local isomorphisms G/H S/T , and under the products of Lie subsupergroups M H U .The classical counterparts of these formulae have numerous applications in harmonic analysis.