  The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional
analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the
space $\Xi$ of univalent functions is an analog of the corresponding classical
complex Cartan domains. We present explicit formulas for realizations of
highest weight representations of $Diff$ in the space of holomorphic
functionals on $\Xi$, reproducing kernels on $\Xi$ determining inner products,
and expressions ('canonical cocycles') replacing spherical functions.
