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

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.