Tuesday 17 June 2025, 16:00 - 17:00 in HG00.062
Tobias Simon (FAU)
Growth of holomorphic extensions of orbit maps of K-finite vectors at the boundary of the crown domain
For a semisimple Lie group G with global Cartan decomposition G=K\exp(\p), there exists a complex domain - we call the principal crown bundle - containing G as totally real submanifold to which orbit maps of K-finite vectors in globalizations of Harish-Chandra modules extend holomorphically. This Extension Theorem is a result by Krötz and Stanton. A natural question in this context is the asymptotic behaviour of these orbit map as one approaches the boundary of the principal crown bundle. In this talk, we sketch how one can obtain estimates for the growth at the boundary with the growth. The key tool is Casselman's Subrepresentation Theorem, which gives models for the representation as subrepresentations of principal series representations, and then control the growth of the Iwasawa component maps in the imaginary direction in a uniform way. As an application of our growth estimates one obtains that the orbits maps have boundary values in the distribution vectors of the representation.