Unramified Whittaker functions for \(GL(3,\mathbb{R})\)
type: Item
instance of: scholarly article
title: Unramified Whittaker functions for \(GL(3,\mathbb{R})\)
zbMATH Open document ID: 0839.22016
DOI: 10.1007/BF02788764
author: Yanyan Li
published in: Journal d'Analyse Mathématique
publication date: 1996-06-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 33C80
zbMATH DE Number: 807399
zbMATH Keywords: Whittaker functions
zbMATH Keywords: unramified principal series representations
zbMATH Keywords: diagonal matrices
zbMATH Keywords: eigenfunctions
zbMATH Keywords: differential operators
zbMATH Keywords: integral representation
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Automorphic forms on GL(3,\({\mathbb{R}})\)
cites work: Q3820683
cites work: On Mellin transforms of unramified Whittaker functions on GL(3,C)
cites work: Fonctions de Whittaker associées aux groupes de Chevalley
cites work: Q4059363
cites work: The multiplicity one theorem for \(\mathrm{GL}_n\)
cites work: Poincaré series for GL(3,\({\mathbb{R}})\)-Whittaker functions
cites work: On explicit integral formulas for \(\text{GL}(n, \mathbb R)\)-Whittaker functions. With an appendix by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein
cites work: Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem
cites work: Q3346501
cites work: Q5846799
full work available at URL: https://doi.org/10.1007/bf02788764
OpenAlex ID: W1979513726
Recommended article: On Mellin transforms of unramified Whittaker functions on GL(3,C)
Recommended article: The archimedean Whittaker functions on GL(3)
Recommended article: Unramified Whittaker functions for certain Brylinski-Deligne covering groups
Recommended article: Unramified Whittaker functions on p-adic loop groups
Recommended article: Unramified Whittaker Functions on the Metaplectic Group
Recommended article: Poincaré series for GL(3,\({\mathbb{R}})\)-Whittaker functions
Recommended article: Q3006800
Recommended article: Whittaker functions for generalized principal series representations of \(\mathrm{SL}(3, \mathbb R)\)
Recommended article: Calculus of principal series Whittaker functions on \(\text{GL}(3,\mathbb C)\)
Recommended article: New formulas for Whittaker functions on \(\text{GL}(n,\mathbb R)\)
description: scientific article; zbMATH DE number 807399