Growth of Taylor coefficients over complex homogeneous spaces (Q611823)

This is a detailed paper devoted to the properties of the so-called Taylor map (that sends a holomorphic function to its set of Taylor coefficients). It is shown that such a map is unitarian as an operator sending the Hilbert space of holomorphic functions on a complex Lie group onto a special Hilbert space contained in the dual of the universal enveloping algebra. As a result, in particular, the authors obtain the first example of unitarity of the Taylor map for a complex manifold which is not a Lie group. The paper is based on previous results of the authors' paper [\textit{B. K. Driver, L. Gross} and \textit{L. Saloff-Coste}, J. Eur. Math. Soc. (JEMS) 11, No. 5, 941--978 (2009; Zbl 1193.32023)], which are used both in the setting up of the problems and in the constructions.