Mellin transforms of Whittaker functions on \(\text{GL} (4,\mathbb{R})\) and \(\text{GL} (4,\mathbb{C})\)
type: Item
instance of: scholarly article
title: Mellin transforms of Whittaker functions on \(\text{GL} (4,\mathbb{R})\) and \(\text{GL} (4,\mathbb{C})\)
zbMATH Open document ID: 0842.11019
DOI: 10.1007/BF02570491
author: Eric Stade
published in: Manuscripta Mathematica
publication date: 1996-01-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156104
review text: The author considers the Whittaker functionals associated with representations of \(\text{GL} (4, \mathbb{R})\) and \(\text{GL} (4, \mathbb{C})\), or rather their Mellin transforms. First of all he shows how the complex case reduces simply to the real one. The Mellin transform in the real case can be found in terms of the generalized hypergeometric functions of W. N. Bailey and L. J. Slater, more precisely, by a well-poised \({}_7 F_6\) function evaluated at 1. The author also proves a new formula in the theory of these functions. The formula for the Mellin transform is unexpected and interesting; it opens a number of further problems, both in itself and through the interactions with other investigations; these are discussed in the final section of this paper.
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 824176
zbMATH Keywords: Whittaker functionals
zbMATH Keywords: representations
zbMATH Keywords: Mellin transforms
zbMATH Keywords: generalized hypergeometric functions
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Q4760447
cites work: Automorphic forms on GL(3,\({\mathbb{R}})\)
cites work: Barnes' Second Lemma and its Application to Rankin-Selberg Convolutions
cites work: On Mellin transforms of unramified Whittaker functions on GL(3,C)
cites work: Cubic metaplectic forms on GL(3)
cites work: Mellin transforms of Whittaker functions
cites work: Zeta functions of simple algebras
cites work: Q3671491
cites work: Whittaker functions on semisimple Lie groups
cites work: Fonctions de Whittaker associées aux groupes de Chevalley
cites work: On Whittaker vectors and representation theory
cites work: Q4095066
cites work: Q3237356
cites work: The multiplicity one theorem for \(\mathrm{GL}_n\)
cites work: Q5508372
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: GL(4, ℝ)-Whittaker Functions and 4 F 3 (1) Hypergeometric Series
cites work: Hypergeometric Series and Euler Factors at Infinity for L-Functions on GL(3, R )×GL(3, R )
cites work: Q4527707
cites work: Q3346501
OpenAlex ID: W2048705012
description: scientific article; zbMATH DE number 824176