Odd orbits in Lie superalgebras

I will present the results of a work with progress joint with V. Serganova. Given a supergroup G whose even subgroup G_0 is reductive, we consider the G_0 orbits of odd elements in the Lie superalgebra Lie(G).  I will describe several flavours of nilpotency in this setting, and how each odd element in the Lie superalgebra defines a ``cohomological'' functor on the representations of G. No previous knowledge of superalgebras is assumed.