Certain L^2-norm and Asymptotic bounds of Whittaker Function for GL(n)

TL;DR

This paper establishes L^2-norm bounds and asymptotic behavior of Whittaker functions on GL(n,R), extending previous integrability results and providing new bounds for principal series representations.

Contribution

It introduces new L^2-norm bounds and asymptotic estimates for Whittaker functions on GL(n,R), extending classical results and analyzing their behavior over the entire group.

Findings

Proved square integrability of Whittaker functions with respect to specific measures.

Derived various asymptotic bounds for smooth Whittaker functions on GL(n,R).

Extended the theorem of Jacquet and Shalika to broader contexts.

Abstract

Whittaker functions of $GL(n, \mathbb R)$ , are most known for its role in the Fourier-Whittaker expansion of cusp forms. Their behavior in the Siegel set, in large, is well-understood. In this paper, we insert into the literature some potentially useful properties of Whittaker function over the group $GL(n, \mathbb R)$ and the mirobolic group $P_n$. We proved the square integrabilty of the Whittaker functions with respect to certain measures, extending a theorem of Jacquet and Shalika . For principal series representations, we gave various asymptotic bounds of smooth Whittaker functions over the whole group $GL(n, \mathbb R)$. Due to the lack of good terminology, we use whittaker functions to refer to $K$-finite or smooth vectors in the Whittaker model.