  With the usual definition of a super Hilbert space and a super unitary
representation, it is easy to show that there are lots of super Lie groups for
which the left-regular representation is not super unitary. I will argue that
weakening the definition of a super Hilbert space (by allowing the super scalar
product to be non-homogeneous, not just even) will allow the left-regular
representation of all (connected) super Lie groups to be super unitary (with an
adapted definition). Along the way I will introduce a (super) metric on a
supermanifold that will allow me to define super and non-super scalar products
on function spaces and I will show that the former are intimately related to
the Hodge-star operation and the Fermionic Fourier transform. The latter also
allows me to decompose certain super unitary representations as a direct
integral over odd parameters of a family of super unitary representations
depending on these odd parameters.
