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

Neretin Yury

doi:10.3934/jgm.2017009

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2017, Volume 9, Issue 2: 207-225. Doi: 10.3934/jgm.2017009

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

Neretin Yury^1,2,3,

1.
Math. Dept., University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria
2.
Institute for Theoretical and Experimental Physics (Moscow), Russia
3.
MechMath. Dept., Moscow State University, Institute for Information Transmission (Moscow), Russia

* Corresponding author: Yury Neretin
* Corresponding author: Yury Neretin

Received: November 30, 2015

Revised: July 31, 2016

Published: October 2017

Supported by FWF grants P25142, P28421

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Abstract

The group $\text{Diff}\left( {{S}^{1}} \right)$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $\text{U}(p,q)$, $\text{Sp}(2n,\mathbb{R})$, $\text{SO}^*(2n)$; 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 $\text{Diff}\left( {{S}^{1}} \right)$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.

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Mathematics Subject Classification: Primary: 81R10, 46E22; Secondary: 30C99, 43A90.

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