In this talk, we'll report on progress towards a conjecture of
B.Krötz about the holomorphic extensions of  non-zero K-finite vectors of
irreducible admissible Banach representations of simple real Lie groups and
the relation to a distinguished domain - the so-called crown domain. We'll
explain some of the main ideas - the Casselman-Wallach smooth
globalisation, vanishing of matrix coefficients at infinity etc. Indeed we
prove the conjecture with some additional growth conditions on the Banach
globalisations. This is joint work with Gang Liu, Uni. Metz.