Poincaré series for GL(n): Fourier expansion, Kloosterman sums, and algebreo-geometric estimates
type: Item
instance of: scholarly article
title: Poincaré series for GL(n): Fourier expansion, Kloosterman sums, and algebreo-geometric estimates
zbMATH Open document ID: 0612.10020
DOI: 10.1007/BF01163653
author: Solomon Friedberg
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173873
review text: In this paper GL(n) Poincaré series are introduced, and their (Piatetski-Shapiro-Shalika) Fourier expansions are computed. These are shown to involve certain exponential sums, which are GL(n) analogues of the classical Kloosterman sum; one such sum arises for each element of the Weyl group W of GL(n). If \(\Gamma =SL(n, {\mathbb{Z}})\) and \(\Gamma _{\infty}\) is the subgroup of upper triangular unipotents of \(\Gamma\), then these sums are given as character sums over a certain quotient space of \(\Gamma _{\infty}\setminus \Gamma\). A coordinatization map is introduced which allows one to realize this space as a subset of the integral points of an affine algebraic variety. This is used to obtain an algebro-geometric estimate for the Kloosterman sum corresponding to \(w=(I_{n-1}^ 1)\) in W (I \(= identity)\), which generalizes the GL(2) majorization of Weil.
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 3989448
zbMATH Keywords: GL(n) Poincaré series
zbMATH Keywords: Fourier expansions
zbMATH Keywords: exponential sums
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: Weyl group
zbMATH Keywords: character sums
MaRDI profile type: Publication
cites work: Q5607044
cites work: Q5611784
cites work: Automorphic forms on GL(3,\({\mathbb{R}})\)
cites work: Poincaré series and Kloosterman sums for SL(3, Z)
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Sums of Kloosterman sums
cites work: Q5517939
cites work: The multiplicity one theorem for \(\mathrm{GL}_n\)
cites work: Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem
cites work: On Some Exponential Sums
cites work: Poincaré series on \(\mathrm{GL}(r)\) and Kloosterman sums
description: scientific article; zbMATH DE number 3989448