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Title:
Localization of integrals on supermanifolds with application to representation theory of supergroups.
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Abstract:
We compute volumes of supergrassmannians and odd symmetric grassmannians using Schwarz-Zaboronsky localization formula which expresses a Berezin integral as a sum of local contributions at all singular points of an odd vector field. To generalize this computation to other classical supermanifolds, we need a CS analogue of localization, since manyclassical supergroups don't have compact real forms. We prove an analogue of the Schwarz-Zaboronsky localization formula for complex smooth supermanifolds. Let X be a compact CS manifold and Q an odd vector field on X such that [Q,Q] is compact. Assume that Q has isolated singular points on X and preserves a volume form w. Then the integral of w over X equals the sum of local contribution at all singular points. We apply the localization formula in the case of homogeneous superspace X=G/K which admits a G invariant volume form. For specific choices of G and K we show that the integral of w over X is not zero. This allows us to use the unitary trick and show that K is a splitting subgroup of G, i.e. the restriction functor from Rep G to Rep K induces injection of Ext groups. In particular, we prove that a defect subgroup is splitting in the case when Lie G is any basic classical or exceptional superalgebra. This has several applications in support theory for supergroups. For example, we show that the DS associated variety detects projectivity in the category of finite-dimensional G-modules. The talk is based on joint papers with A. Sherman and D. Vaintrob.
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