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

TL;DR

This paper explores the representation theory of the infinite-dimensional diffeomorphism group of the circle, providing explicit formulas for highest weight representations, reproducing kernels, and analogs of spherical functions.

Contribution

It introduces explicit realizations of highest weight representations and reproducing kernels for the diffeomorphism group of the circle, extending classical Lie group concepts to an infinite-dimensional setting.

Findings

Explicit formulas for highest weight representations

Reproducing kernels for the space of univalent functions

Canonical cocycles as 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.