# Invariant Berezin integration on homogeneous supermanifolds

## TL;DR

This paper investigates the conditions for the existence of invariant Berezinian sections on homogeneous supermanifolds and derives integral formulas analogous to classical harmonic analysis results.

## Contribution

It introduces a method to express the Berezinian line bundle as an associated bundle and derives Fubini-type formulas for invariant integration on supermanifolds.

## Key findings

- Established criteria for unimodularity of G/H supermanifolds.
- Derived a Fubini-type formula for invariant integrals.
- Extended integral transformation formulas to supermanifolds.

## Abstract

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 over G/H. We also study the fibre integration of Berezinians on oriented fibre bundles. As an application, we prove a formula of `Fubini' type: the invariant integral over G can be expressed (up to sign) by a succesive invariant integration over H and G/H. Moreover, we derive analogues of integral formulae for the transformation under local isomorphisms of unimodular homogeneous superspaces G/H and S/T, and under the products of subsupergroups of Lie supergroups. The classical counterparts of these formulae have numerous applications in harmonic analysis.