Representations of super Lie groups: differentiable representations,K-finite and analytic vectors

It is well-known that constructing unitary representations of super Lie groups is a non trivial problem. It seems reasonable to approach the representation theory of super Lie groups from a more general perspective namely that of representations on super Frechet spaces. The basic tools needed in this approach are the smooth and analytic vectors, and, in particular, the subspace of the analytic vectors whose elements  are the K-finite. We review this concepts and we outline a possible generalization to the super setting.