Invariant Berezin integration on homogeneous supermanifolds

Details Invariant Berezin integration on homogeneous supermanifolds Invariant Berezin integration on homogeneous supermanifolds

2009-11-17

arxiv.org/abs/0911.3251v1

J. Lie Theory 20 (2010) no. 1 65-91

Mathematics

Differential Geometry

26 pages

Scientific paper

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.

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