The group of diffeomorphisms of the circle: reproducing kernels and analogs of spherical functions - Neretin, Yury A (arXiv.org, 2016)
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 \(Ξ\) 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 \(Ξ\), reproducing kernels on \(Ξ\) determining inner products, and expressions ('canonical cocycles') replacing spherical functions.