Title: Asymptotic behaviour of (Z(\g),K)-finite smooth functions of moderate growth at the boundary of the complex crown domain
Abstract: By results from Krötz and Schlichtkrull every K-finite smooth joint eigenfunction on a connected semisimple Lie group
extends to a holomorphic function on the universal covering of the principal K_\C-fibre bundle over the complex crown domain.
A natural way to see this is by writing the K-finite smooth joint eigenfunction as a vector-valued Poisson transform, i.e. applying
a hyperfunction vector to the orbit map of a K-finite vector. If the hyperfunction vector is a distribution vector, i.e. the function is
of moderate growth, we prove that the asymptotic at the boundary of the complex crown domain is at most polynomial. Using
these growth rates, one obtains estimates for continuous seminorms in smooth Frechet globalizations of moderate growth of
Harish-Chandra modules of finite length.
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