Smooth transfer of Kloosterman integrals (the Archimedean case)

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ 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 ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas of this work. However we have to face additional difficulties that appear only in the Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.