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

  • * Corresponding author: Yury Neretin

    * Corresponding author: Yury Neretin

Supported by FWF grants P25142, P28421

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  • 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.

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


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