On explicit integral formulas for \(\text{GL}(n, \mathbb R)\)-Whittaker functions. With an appendix by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein
type: Item
instance of: scholarly article
title: On explicit integral formulas for \(\text{GL}(n, \mathbb R)\)-Whittaker functions. With an appendix by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein
zbMATH Open document ID: 0731.11027
DOI: 10.1215/S0012-7094-90-06013-2
author: Eric Stade
published in: Duke Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The author considers the Whittaker functions \(W_{(n,\nu)}(y)\) associated with nonramified principal series representations of the group \(\text{GL}(n, \mathbb R)\). The main result is: Theorem. Let \(n\geq 2\). If \(\nu \in {\mathbb C}^{n-1}\), put \(\lambda_{j-1}=n\nu_ j/(n-2)\) for \(2\leq j\leq n- 2\) and \(\lambda =(\lambda_ 1,\lambda_ 2,...,\lambda_{n-3})\). Also define \(u_ 0=1/u_{n-1}=0\) and \(u^ 0_{n-1}=1\). Then \[ W^*_{(n,\nu)}(y)=2^{n-1}\int_{({\mathbb R}_+)^{n-2}}\prod^{n- 1}_{i=1}u_ i^{r_{i,1}-r_{i,n-i}}K_{\mu_ 1}(2\pi y_ i\sqrt{(1+u^ 2_{i-1})(1+1/u_ i^ 2)}) \] \[ \times W^*_{(n- 2,\lambda)}(\frac{y_ 2}{u_ 2}u_ 1,\frac{y_ 3}{u_ 3}u_ 2,...,\frac{y_{n-2}}{u_{n-2}}u_{n-3})\prod^{n- 2}_{i=1}\frac{du_ i}{u_ i} \] where \(K\) denotes the \(K\)-Bessel function \(K_{\mu}(2\pi y)=\frac{1}{2}\int^{\infty}_{0}t^{\mu}\exp (-\pi y(t+1/t))\frac{dt}{t}\) \((y>0)\). Here \(W^*_{(n,\nu)}\) is \(W_{(n,\nu)}\) divided by some powers of the \(y_ i\)'s. As applications of the above theorem, he expresses the local factors at the archimedean places for the exterior square automorphic \(L\)-function on \(\text{GL}(n)\) and for automorphic functions for \(\text{GL}(2,\mathbb R)\times \text{GL}(3,\mathbb R)\) as products of Gamma functions. He also gives growth estimates for \(W_{(n,\nu)}\).
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 4208165
zbMATH Keywords: automorphic L-functions
zbMATH Keywords: local factors
zbMATH Keywords: Whittaker functions
zbMATH Keywords: principal series representations
zbMATH Keywords: growth estimates
MaRDI profile type: Publication
cites work: Automorphic forms on GL(3,\({\mathbb{R}})\)
cites work: Q3820683
cites work: Barnes' Second Lemma and its Application to Rankin-Selberg Convolutions
cites work: Q3497144
cites work: Q3671491
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: Poincaré series for GL(3,\({\mathbb{R}})\)-Whittaker functions
cites work: Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem
full work available at URL: s0012-7094-90-06013-2
OpenAlex ID: W1977825208
description: scientific article; zbMATH DE number 4208165