Title: Smooth Transfer of Kloosterman Integrals

Abstract:
Fix a local field $F$ and it's quadratic extension $E$. We consider an integral of a Schwartz function on $\GL_n(F)$ along the orbits of the two sided action of the groups of upper 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 $\GL_n(F)$ is replaced with the variety of non-degenerate hermitian forms.
 
The non-Archimedean case is done by Jaquet and the Archimedean case is done by Gourevitch and myself. I will discuss the main ingredients of the proof, the difficulties that occur in the Archimedean case and the methods we used  to overcome them.