Holomorphic extensions of representations of the group of the diffeomorphisms of the circle

This paper gives the construction of a semigroup $\Gamma$ which could be thought of as the complexincation of the group $\operatorname{Diff}$ of analytic diffeomorphisms of the circle, and it is shown that any unitary projective representation of $\operatorname{Diff}$ with highest weight has a holomorphic extension to $\Gamma$. For this, $\Gamma$ is embedded in the semigroup of “endomorphisms of canonical commutation relations” (this is a certain part of the Lagrange Grassmannian in complex symplectic Hilbert space).
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