Kernels of Langlands automorphic transfer and nonlinear Poisson formulas
type: Item
instance of: scholarly article
title: Kernels of Langlands automorphic transfer and nonlinear Poisson formulas
zbMATH Open document ID: 1300.11033
author: Laurent Lafforgue
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2014-06-18T00:00:00Z
review text: The subject of this article is the conjectural automorphic transfer from a quasi-split reductive group \(G\) to a general linear group \(\mathrm{GL}_r\) corresponding to a homomorphism \(\rho\) from the L-group of \(G\) to \(\mathrm{GL}_r({\mathbb C})\). The author introduces the notion of kernel for this transfer. A kernel is an automorphic function on \(G({\mathbb A}) \times G({\mathbb A}) \times \mathrm{GL}_r({\mathbb A})\) which satisfies a condition regarding the action of the spherical local Hecke algebras of \(G\) and \(\mathrm{GL}_r\) at the unramified places, and an integral with this kernel defines a linear transformation from automorphic functions on \(G({\mathbb A})\) to automorphic functions on \(G({\mathbb A}) \times \mathrm{GL}_r({\mathbb A})\). The construction of sufficiently many kernels for \(\rho\) will imply Langlands transfer. In the present article \(G\) is assumed to be defined over a function field \(F\). The construction of kernels goes via the construction of their Fourier coefficients, which leads to the definition of local kernels at the unramified places. For each place \(x\) a local \(\rho\)-Fourier transform is defined on a certain function space on \(G(F_x)\) and the product over all places defines the global \(\rho\)-Fourier transform. When the Langlands conjecture on transfer by \(\rho\) is assumed, it is possible to derive a Poisson formula for the \(\rho\)-Fourier transform, valid for global functions which are highly ramified at at least one place (``Poisson formula without boundary terms''). Conversely, when the Poisson formula without boundary terms on \(G({\mathbb A}) \times \mathrm{GL}_{r-1}({\mathbb A})\) is known, it is possible to construct sufficiently many kernels for \(G\) and \(\rho\) and thus prove the transfer by \(\rho\). Here it is necessary to consider also the ramified places. The techniques used are clearly explained (spectral decomposition of Whittaker functions, etc.). It is expected that the Poisson formula …
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6305054
zbMATH Keywords: automorphic functions
zbMATH Keywords: representations over local fields
zbMATH Keywords: representations over adelic rings
zbMATH Keywords: adelic Poisson formulas
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: s11537-014-1274-y
OpenAlex ID: W2045264840
cites work: Converse theorems for \(\text{GL}_n\)
cites work: Zeta functions of simple algebras
cites work: Rankin-Selberg Convolutions
cites work: A lemma on highly ramified \(\epsilon\)-factors
DOI: 10.1007/S11537-014-1274-Y
description: scientific article; zbMATH DE number 6305054