Invariant Berezin Integration on Homogeneous Supermanifolds

Alexander Alldridge¹ ; Joachim Hilgert¹

¹Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33100 Paderborn, Germany

Journal of Lie Theory, Tome 20 (2010) no. 1, pp. 65-91

Résumé

\def\g{{\frak g}} \def\h{{\frak h}} Let $\cal G$ be a Lie supergroup and $\cal H$ a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold $\cal G/\cal H$, i.\ e.\ the existence of $\cal G$-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a $\cal G$-equivariant associated bundle of the principal $\cal H$-bundle $\cal G\to \cal G/\cal H$. We also study the fibre integration of Berezinians on oriented fibre bundles. As an application, we prove a formula of `Fubini' type: $$ \int_{\cal G}f = (-1)^{\dim\h_1\cdot\dim\g/\h}\int_{\cal G/\cal H} \int_{\cal H}f,\ \text{for all}\ f\in\Gamma_c(G,\cal O_{\cal G}). $$ Moreover, we derive analogues of integral formulae for the transformation under local isomorphisms $\cal G/\cal H\to \cal S/\cal T\!$, and under the products of Lie subsupergroups $\cal M\cdot\cal H\subset\cal U$. The classical counterparts of these formulae have numerous applications in harmonic analysis.

Détail

Classification : 58A50, 58C50, 53C30
Mots-clés : Supermanifold, Lie supergroup, homogeneous superspace, Berezin integral, invariant Berezinian form, unimodularity, Fubini formula, fibre integration