Unramified Whittaker functions for \(GL(3,\mathbb{R})\)
The Whittaker functions considered in this paper are those associated with unramified principal series representations of \(\text{GL} (3, \mathbb{R})\). They are determined by their values on diagonal matrices of the form \(\text{diag} (y_1 y_2, y_1, 1)\). For a fixed additive character these functions are eigenfunctions of two differential operators and the first author has shown that these equations have, in general, six linearly independent solutions. One of these, denoted by \(W_\lambda (y_1, y_2)\), is characterized by being the smallest at infinity. The purpose of this paper is to derive the precise asymptotic development of \(W_\lambda (y_1, y_2)\) as \(y_1 \to \infty\) and \(y_1 y_2^\alpha\) remains constant. This is done using the saddle-point method applied to an integral representation of \(W_\lambda (y_1, y_2)\). The authors give an algorithm for computing any level of accuracy in the asymptotic development. They also summarize a great deal of other information about this class of functions.
unramified principal series representations
scientific article; zbMATH DE number 807399
scientific article