The group of diffeomorphisms of the circle: reproducing kernels and analogs of spherical functions

Abstract

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.