Smooth Transfer (the Archimedean case)

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.

24 pages, no figures

References in corpus (3)

• Schwartz functions on Nash manifolds
• Generalized Harish-Chandra descent, Gelfand pairs and an Archimedean analog of Jacquet-Rallis' Theorem
• De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

Cited by in corpus (5)

• The Schwartz space of a smooth semi-algebraic stack
• Twisted homology for the mirabolic nilradical
• Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture
• Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices
• Generalized Whittaker quotients of Schwartz functions on G-spaces