Kernels of Langlands automorphic transfer and nonlinear Poisson formulas
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 without boundary terms can be completed with additional terms to a Poisson formula valid for global functions which are not necessarily highly ramified at some place. The last chapter sketches a way how this possibly can be done.
scientific article; zbMATH DE number 6305054