Localization in supergeometry, and towards a Green's correspondence for supergroups


A smooth supervariety is (roughly speaking) the exterior algebra of a vector bundle on a smooth variety.  Odd vector fields on supervarieties often encode important complexes of the underlying variety, such as a Koszul complex or the de Rham complex.  On the other hand, computing the cohomology of more general odd vector fields is an important problem in the representation theory of supergroups, as well as physics.  We explain a recent localization theorem which computes in many cases the cohomology of an odd vector field, using our understanding of Koszul complexes in commutative algebra.  Always close at hand will be the example of affine superspace, to illustrate phenomena in a gentle setting.  We then look to present an application: we'll first recall (roughly) what Green's correspondence says for finite groups in modular representation theory, and then explain how the localization theorem gives us a first step towards this correspondence in the super setting.  Part of a joint work with I. Entova-Aizenbud and V. Serganova.