$K$-finite Whittaker functions are of finite order one
Volume 158 / 2013
Acta Arithmetica 158 (2013), 359-401
MSC: Primary 11F70, 22E45, 33C15; Secondary 11M99, 30D15.
DOI: 10.4064/aa158-4-4
Abstract
We prove a finite order one type estimate for the Whittaker function attached to a $K$-finite section of a principle series representation of a real or complex Chevalley group. Effective computations are made using convexity in $\mathbb {C}^n$, following the original paper of Jacquet. As an application, we give a simplified proof of the known result of the boundedness in vertical strips of certain automorphic $L$-functions, using a result of Müller.
