Polynomial growth of holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown

Abstract

The Kr\"otz-Stanton Extension Theorem states that the orbit map of a K-finite vector in a Hilbert representation of a linear Lie group extends to a holomorphic map to a principal fibre bundle over the complex crown domain associated to the Riemannian symmetric space $G/K$. We extend this theorem to arbitrary connected semisimple Lie groups and prove polynomial growth estimates at the boundary. Using this, we show that the boundary values of these holomorphic extensions exist in the space of distribution vectors.