Smooth Transfer (the Archimedean case)

TL;DR

This paper constructs a transfer between Schwartz functions on GL(n,R) and non-degenerate Hermitian forms, compatible with Kloosterman integrals, extending known non-Archimedean results to the Archimedean setting.

Contribution

It establishes the existence of a smooth transfer compatible with Kloosterman integrals in the Archimedean case, addressing unique challenges not present in non-Archimedean scenarios.

Findings

Proves the transfer coincides with the non-degenerate Hermitian forms case.

Extends Jacquet's non-Archimedean results to Archimedean setting.

Addresses additional difficulties specific to Archimedean analysis.

Abstract

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on GL(n,R) and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on GL(n,R) along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non-degenerate character. This gives a smooth function on the torus. We prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when GL(n,R) is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for gl(n,R). The non-Archimedean case is done by Jacquet and our proof follows the same lines. However we have to face additional essential difficulties that appear only in the Archimedean case.